A series can be represented in a compact form, called summation or sigma notation. The Greek capital letter, **∑**, is used to represent the sum. The series 4+8+12+16+20+24 can be expressed as 6∑n=14n. The expression is read as the sum of 4n as n goes from 1 to 6.

Contents

- 1 How do you use summation?
- 2 How do you find the summation?
- 3 How do you do infinity on a calculator?
- 4 Finding Sums of Finite Arithmetic Series – Sequences and Series (Algebra 2)
- 5 Summation Notation
- 6 Arithmetic Series
- 7 Arithmetic Sequences and Sums
- 8 Arithmetic Sequence
- 9 Advanced Topic: Summing an Arithmetic Series
- 10 Footnote: Why Does the Formula Work?
- 11 Sigma Notation
- 12 More Powerful
- 13 Why is it called “Sigma”
- 14 Sigma Notation and Series – MathBitsNotebook(A2
- 15 Summation Notation
- 16 Problems
- 17 9.2: Summation Notation
- 18 Contributors and Attributions
- 19 Sigma notation for sums – Topics in precalculus
- 20 Series and Their Notations
- 21 Arithmetic Series
- 21.1 A General Note: Summation Notation
- 21.2 QA
- 21.3 How To: Given summation notation for a series, evaluate the value.
- 21.4 Example: Using Summation Notation
- 21.5 Try It
- 21.6 Arithmetic Series
- 21.7 A General Note: Formula for the Partial Sum of an Arithmetic Series
- 21.8 How To: Given terms of an arithmetic series, find the partial sum
- 21.9 Example: Finding the partial sum of an Arithmetic Series
- 21.10 Try It
- 21.11 Example: Solving Application Problems with Arithmetic Series
- 21.12 Try It

- 22 Geometric Series
- 22.1 A General Note: Formula for the Sum of the FirstnTerms of a Geometric Series
- 22.2 How To: Given a geometric series, find the sum of the firstnterms.
- 22.3 Example: Finding the FirstnTerms of a Geometric Series
- 22.4 Try It
- 22.5 Example: Solving an Application Problem with a Geometric Series
- 22.6 Try It

- 23 Using the Formula for the Sum of an Infinite Geometric Series
- 23.1 Determining Whether the Sum of an Infinite Geometric Series is Defined
- 23.2 DETERMINING WHETHER THE SUM OF AN INFINITE GEOMETRIC SERIES IS DEFINED
- 23.3 How To:Given the first several terms of an infinite series, determine if the sum of the series exists.
- 23.4 Example: Determining Whether the Sum of an Infinite Series is Defined
- 23.5 try it

- 24 Finding Sums of Infinite Series
- 25 Annuities
- 26 Key Equations
- 27 Key Concepts
- 28 Glossary

## How do you use summation?

A series can be represented in a compact form, called summation or sigma notation. The Greek capital letter, ∑, is used to represent the sum. The series 4+8+12+16+20+24 can be expressed as 6∑n=14n. The expression is read as the sum of 4n as n goes from 1 to 6.

## How do you find the summation?

To find the sum of the natural numbers from 1 to n, we use the formula n (n + 1) / 2. For example, the sum of the first 50 natural numbers is, 50 (50 + 1) / 2 = 1275.

## How do you do infinity on a calculator?

Example – To specify positive infinity, input 1E99. To specify negative infinity, input -1E99. The “E” symbol is short for scientific notation and can be accessed by pressing [2nd] [EE]. -1E99 can be interpreted as “negative one times 10 to the ninety-ninth power”.

## Finding Sums of Finite Arithmetic Series – Sequences and Series (Algebra 2)

The sum of all terms for a finitearithmetic sequencegiven bywherea1 is the first term,dis the common difference, andnis the number of terms may be determined using the following formula: wherea1 is the first term,dis the common difference, andnis the number of terms Consider the arithmetic sum as an example of how the total is computed in this manner. If you choose not to add all of the words at once, keep in mind that the first and last terms may be recast as2fives. This method may be used to rewrite the second and second-to-last terms as well.

There are 9fives in all, and the aggregate is 9 x 5 = 9.

expand more Due to the fact that the difference between each term is constant, this sequence from 1 through 1000 is arithmetic.

The first phrase, a1, is one and the last term, is one thousand thousand.

The sum of all positive integers up to and including 1000 is 500 500.

## Summation Notation

The summation notation is a straightforward means of showing the sum of a limited (and ultimately terminating) number of terms in a series. This is represented by the Greek symbol sigma, which means “sigma.” When employing the sigma notation, the variable specified below the sigma symbol is referred to as the index of summation (or index of addition). Specifically, the lower number represents the lower limit of the index (the term at which the summing begins), while the higher number represents the top limit of the summation (the term where the summation ends).

##### Example 1

Make a list of the terms for the following sums, and then compute the total.

##### Example 2

Each series should be expressed in sigma notation.

- 8 plus 11 plus 14 plus 17 plus 20 = This is an arithmetic series of five terms, the first of which is 8 and the last of which is 3, with the common difference being 3. As a result, a1= 8 and d= 3. The next phrase in the matching series is the Because there are five words in the series, it may be written as follows: This is a geometric series with six terms, the first of which is and the last of which is, and whose common ratio is. Therefore,and. In that case, the th term of the matching sequence is used. Due to the fact that there are six terms in the given series, the total may be expressed as

## Arithmetic Series

It is the sum of the terms of an arithmetic sequence that is known as an arithmetic series. A geometric series is made up of the terms of a geometric sequence and is represented by the symbol. You can work with other sorts of series as well, but you won’t have much experience with them until you get to calculus. For the time being, you’ll most likely be collaborating with these two. How to deal with arithmetic series is explained and shown on this page, among other things. You can only take the “partial” sum of an arithmetic series for a variety of reasons that will be explored in greater detail later in calculus.

The following is the formula for the firstnterms of the anarithmeticsequence, starting with i= 1, and it is written: Content Continues Below The “2” on the right-hand side of the “equals” sign may be converted to a one-half multiplied on the parenthesis, which reveals that the formula for the total is, in effect,n times the “average” of the first and final terms, as seen in the example below.

The summation formula may be demonstrated via induction, by the way. The sum of the firstnterms of a series is referred to as the “then-th partial sum,” and it is sometimes symbolized by the symbol “Sn.”

#### Find the35 th partial sum,S 35, of the arithmetic sequence with terms

The first thirty-five terms of this sequence are added together to provide the 35th partial sum of the series. The first few words in the sequence are as follows: Due to the fact that all of the words share a common difference, this is in fact an arithmetic sequence. The final term in the partial sum will be as follows: Plugging this into the formula, the 35 th partial sum is:Then my answer is:35 th partial sum:Then my answer is:35 th partial sum: S 35 = 350 S 35 = 350 If I had merely looked at the formula for the terms in the series above, I would have seen the common difference in the above sequence.

If we had used a continuous variable, such as the “x” we used when graphing straight lines, rather than a discrete variable, then ” ” would have been a straight line that rose by one-half at each step, rather than the discrete variable.

#### Find the value of the following summation:

It appears that each term will be two units more in size than the preceding term based on the formula ” 2 n– 5 ” for the then-thirteenth term. (Whether I wasn’t sure about something, I could always plug in some values to see if they were correct.) As a result, this is a purely arithmetic sum. However, this summation begins at n= 15, not at n= 1, and the summation formula is only applicable to sums that begin at n=1. So, how am I supposed to proceed with this summation? By employing a simple trick: The simplest approach to get the value of this sum is to first calculate the 14th and 47th partial sums, and then subtract the 14th from the 47th partial sum.

By doing this subtraction, I will have subtracted the first through fourteenth terms from the first through forty-seventh terms, and I will be left with the total of the fifteenth through forty-seventh terms, as shown in the following table.

These are the fourteenth and forty-seventh words, respectively, that are required: a14= 2(14) – 5 = 23a47= 2(47) – 5 = 89a14= 2(14) – 5 = 23a47= 2(47) – 5 = 89 With these numbers, I now have everything I need to get the two partial sums for my subtraction, which are as follows: I got the following result after subtracting: Then here’s what I’d say: As a side note, this subtraction may also be written as ” S 47 – S 14 “.

Don’t be shocked if you come into an exercise that use this notation and requires you to decipher its meaning before you can proceed with your calculations; this is common.

If you’re working with anything more complicated, though, it may be important to group symbols together in order to make the meaning more obvious. In order to do so correctly, the author of the previous exercise should have structured the summation using grouping symbols in the manner shown below:

#### Find the value ofnfor which the following equation is true:

Knowing that the first term has the value a1= 0.25(1) + 2 = 2.25, I may proceed to the second term. It appears from the formula that each term will be 0.25 units larger than the preceding term, indicating that this is an arithmetical series withd= 0.25, as shown in the diagram. The summation formula for arithmetical series therefore provides me with the following results: The number n is equal to 2.25 + 0.25 + 2 = 42n is equal to 0.25 + 4.25 + 42 = 420.25 n2+ 4.25 n– 42 = 0n2+ 17 n– 168 = 0(n+ 24)(n– 7 = 0n2+ 17 n– 168 = 0(n+ 24)(n– 7).

Then n= 7 is the answer.

However, your instructor may easily assign you a summation that needs you to use, say, eighty-six words or a thousand terms in order to arrive at the correct total.

As a result, be certain that you are able to do the calculations from the formula.

#### Find the sum of1 + 5 + 9 +. + 49 + 53

Knowing that the first term has the value_a1= 0.25(1) + 2 = 2.25, I know that the second term has the value: The formula indicates that each term will be 0.25 units larger than the preceding term, indicating that this is an arithmetical series with d= 0.25 units. The summation formula for arithmetical series then provides me with the following results. The number n is equal to 2.25 + 0.25 + 2 = 42n is equal to 4.25 + 0.25 + 42 = 420.25 n2+ 4.25 n– 42 = 0n2+ 17 n– 168 = 0(n+ 24)(n– 7 = 0n2+ 17 n– 168 = 0(n+ 24)(n– 7).

In this case, n=7 You might complete the preceding task by adding terms one at a time until you reach the necessary number of 21 words.

The time it would take you to complete each step would be a complete waste of your time, especially on a test.

## Arithmetic Sequences and Sums

A sequence is a collection of items (typically numbers) that are arranged in a specific order. Each number in the sequence is referred to as aterm (or “element” or “member” in certain cases); for additional information, see Sequences and Series.

## Arithmetic Sequence

An Arithmetic Sequence is characterized by the fact that the difference between one term and the next is a constant. In other words, we just increase the value by the same amount each time. endlessly.

### Example:

1, 4, 7, 10, 13, 16, 19, 22, and 25 are the numbers 1 through 25.

Each number in this series has a three-digit gap between them. Each time the pattern is repeated, the last number is increased by three, as seen below: As a general rule, we could write an arithmetic series along the lines of

- There are two words: Ais the first term, and dis is the difference between the two terms (sometimes known as the “common difference”).

### Example: (continued)

1, 4, 7, 10, 13, 16, 19, 22, and 25 are the numbers 1 through 25. Has:

- In this equation, A = 1 represents the first term, while d = 3 represents the “common difference” between terms.

And this is what we get:

### Rule

It is possible to define an Arithmetic Sequence as a rule:x n= a + d(n1) (We use “n1” since it is not used in the first term of the sequence).

### Example: Write a rule, and calculate the 9th term, for this Arithmetic Sequence:

3, 8, 13, 18, 23, 28, 33, and 38 are the numbers three, eight, thirteen, and eighteen. Each number in this sequence has a five-point gap between them. The values ofaanddare as follows:

- A = 3 (the first term)
- D = 5 (the “common difference”)
- A = 3 (the first term).

Making use of the Arithmetic Sequencerule, we can see that_xn= a + d(n1)= 3 + 5(n1)= 3 + 3 + 5n 5 = 5n 2 xn= a + d(n1) = 3 + 3 + 3 + 5n n= 3 + 3 + 3 As a result, the ninth term is:x 9= 5 9 2= 43 Is that what you’re saying? Take a look for yourself! Arithmetic Sequences (also known as Arithmetic Progressions (A.P.’s)) are a type of arithmetic progression.

## Advanced Topic: Summing an Arithmetic Series

To summarize the terms of this arithmetic sequence:a + (a+d) + (a+2d) + (a+3d) + (a+4d) + (a+5d) + (a+6d) + (a+7d) + (a+8d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + (a+9d) + ( make use of the following formula: What exactly is that amusing symbol? It is referred to as The Sigma Notation is a type of notation that is used to represent a sigma function. Additionally, the starting and finishing values are displayed below and above it: “Sum upnwherengoes from 1 to 4,” the text states. 10 is the correct answer.

### Example: Add up the first 10 terms of the arithmetic sequence:

The values ofa,dandnare as follows:

- In this equation, A = 1 represents the first term, d = 3 represents the “common difference” between terms, and n = 10 represents the number of terms to add up.

As a result, the equation becomes:= 5(2+93) = 5(29) = 145 Check it out yourself: why don’t you sum up all of the phrases and see whether it comes out to 145?

## Footnote: Why Does the Formula Work?

Let’s take a look at why the formula works because we’ll be employing an unusual “technique” that’s worth understanding. First, we’ll refer to the entire total as “S”: S = a + (a + d) +. + (a + (n2)d) +(a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n1)d) + (a + (n2)d) + (a + (n1)d) + (a + (n1)d) + (a + (n1)d) + After that, rewrite S in the opposite order: S = (a + (n1)d)+ (a + (n2)d)+. +(a + d)+a. +(a + d)+a. +(a + d)+a. Now, term by phrase, add these two together:

S | = | a | + | (a+d) | + | . | + | (a + (n-2)d) | + | (a + (n-1)d) |

S | = | (a + (n-1)d) | + | (a + (n-2)d) | + | . | + | (a + d) | + | a |

2S | = | (2a + (n-1)d) | + | (2a + (n-1)d) | + | . | + | (2a + (n-1)d) | + | (2a + (n-1)d) |

Each and every term is the same! Furthermore, there are “n” of them. 2S = n (2a + (n1)d) = n (2a + (n1)d) Now, we can simply divide by two to obtain the following result: The function S = (n/2) (2a + (n1)d) is defined as This is the formula we’ve come up with:

## Sigma Notation

This sign (calledSigma) is a mathematical symbol that indicates “to sum up.” Sigma is one of my favorite programs since it is easy to use and can accomplish a variety of interesting things. So that’s a quick summary of the situation.

### Sum What?

Thevalues are shown belowand above the Sigma: | 4Σn=1n | |

it saysngoes from 1 to 4,which is1,2,3and4 | ||

### OK, Let’s Go.

So now we add up 1,2,3 and 4: | 4Σn=1n = 1 + 2 + 3 + 4 =10 |

“add up” is represented by the symbol Sigma. Sigma is one of my favorite programs since it is easy to use and can accomplish a variety of tasks. This is a short summary of the situation.

## More Powerful

But you have the ability to accomplish far more powerful things than that! We may square each time and add the results to get the following: The number n 2 equals one plus two plus three plus four plus five equals thirty. We can put the first four words in this sequence together to get the total. 2n+1: The sum of (2n+1) is 3 + 5 + 7 + 9 = 24. We may also use other letters; for example, we can useiand sum upi (i+1), going from 1 to 3: the product of i(i+1) = 12 plus 23 plus 34 = 20 Furthermore, we can begin and conclude with any number.

Here’s how we move from 3 to 5: A large number of additional examples may be found under the more complex topic of Partial Sums.

## Why is it called “Sigma”

Sigma is the upper case letterSin Greek. AndSstands forS um. |

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## Sigma Notation and Series – MathBitsNotebook(A2

Consider the finite arithmetic sequence 2, 4, 6, 8, 10.Now, consider adding these terms together (taking the sum):2 + 4 + 6 + 8 + 10.Such asequence summationis called aseries,and is designated byS nwherenrepresents the number of terms of the sequence being added.S5= 2 + 4 + 6 + 8 + 10 This course will be dealing withfinite series: sums of a specified number of terms (not infinite sums). S nis often called annthpartial sum, since it can represent the sum of a certain “part” (portion) of a sequence.Apartial sumcustomarily starts witha1and ends witha n, addingnterms. |

Partial Sums: S1= 2S2= 2 + 4S3= 2 + 4 + 6S4= 2 + 4 + 6 + 8S5= 2 + 4 + 6+8 + 10 | S1=a1 S2=a1+a2 S3=a1+a2+a3 S4=a1+a2+a3+ a4 S5=a1+a2+a3+ a4+a5 S n=a1+a2+a3+ a4+a5+. +a n |

Thesummation of a specified number of terms of a sequence (aseries) canalso be represented in a compact form, calledsummation notation, orsigma notation.The Greek capital letter sigma, is used to indicate asum.To write the terms of the series, replacenby the consecutive integersfrom 1 to 5, as shown above. |

Problem: | Solution: |

Evaluate: | Replacejin the expression (j2+ 1) with the values 1, 2, 3 and 4:Notice that the expression (j2+ 1) is placed in a set of parentheses behind the sigma. Without the parentheses, only thej2would be part of the sigma, with the + 1 added on “after” the sigma was completed. |

Evaluate: | Notice that the starting value isi= 2. While the starting value is usually 1, it can actually be any integer value. Also notice how ONLY the variableiis replaced with the values 2, 3, and 4: |

Evaluate: | This is an important pattern strategy to remember!Notice how raising (-1) to a power affected by the signs of the terms in the series. |

Evaluate: | Yes, it is possible to calculate a summation on an expression starting witha negative number. Substitute -2, -1, 0 and 1. Remember, however, that when working with sequences, the lowest starting value is 1. |

Evaluate: | OK, so this is a sneaky one. You know that ln(e x) =x, so this summation is the same aswhich equals1 + 2 + 3 + 4 + 5 = 15. |

6.Use sigma notation to represent3 + 6 + 9 + 12 +.for the first 36 terms. |

Look for a pattern based upon the position of each term. Often making a table will let the pattern to be more easily seen.Sequence formula:a n= 3 n |

7.Use sigma notation to represent-2 + 4 – 6+ 8 – 10+.for the first 100 terms. |

Again, look for a pattern. Remember what we saw in example3 regarding using powers of (-1) to affect the signs of the terms.Sequence formula:a n= (-1)n2 n |

8.Cameron is starting a 6 week jogging program. He will jog8 miles the first week and increase the distance by10% per week. Using sigma notation, write an expression to represent the total number of miles he will have jogged over the 6 week program. | An increase of 10%, is equivalent to 110% per week in the number of miles.Week 1: 8 milesWeek 2:8+.10(8) or 1.10(8) milesWeek 3:1.10(1.10)(8) =(1.10) 2 (8)Week 4:1.10(1.10)(1.10)(8) = (1.10) 3 (8) (and so on.)The pattern is(1.10)n -1 (8). |

Isn’t it interesting to see that the variable used in the summation symbol (sigma) may take on any letter of your choosing? Regardless matter whether variable is utilized, the total will always be the same. When attempting to find an expression for a sequence (series), keep the following strategies in mind:

Series | Possible notation(partial sum) | Strategy to remember |

or | Always remember that there ismore than one possible answer. | |

Patterns can be eitherincreasing or decreasing. | ||

Look to see if a value is beingconsistently added(or subtracted). Arithmetic Sequence | ||

Look to see if a value is beingconsistently multiplied(or divided). Geometric Sequence | ||

Look to see if the values are “famous” numbers such asperfect squares. | ||

Look to see if thesigns alternate. Alternating signs can be handled using powers of -1. |

Partial sums result in the creation of new sequences: Sums (answers) from partial sums of a series may combine to generate a new and intriguing sequence of numbers. Check out some of the partial sums of the summation of positive oddintegers, such as the following:

Partial Sums: S1= 1 = 1S2= 1 + 3 = 4S3= 1 + 3 + 5 = 9S4= 1 + 3 + 5 + 7 = 16S5= 1 + 3 + 5+7 + 9 = 25 | The answers from the partial sums create a sequence ofperfect squares.1, 4, 9, 16, 25 |

For calculatorhelp with Summation Notation (Sigma)Click here. |

## Summation Notation

In many mathematical calculations, the addition of several variables is required. Summation, also known as sigma notation, is a practical and straightforward kind of shorthand that may be used to describe a brief expression for the sum of the values of a variable in a concise manner. The numerals x 1, x 2, x 3,.x n signify a collection of n numbers. x 1 is the first number in the set of four numbers. The ith number in the set is represented by the symbol x i. The following are the components of summation notation: The adverbial summation sign In this example, S represents the Greek upper case letter S, which appears as a symbol.

- To the right of the summation sign is a representation of a typical member of the sequence that is being summed.
- The variable of summation is represented by an index that is put beneath the sign of the summing operation.
- (The letters j and t are also often used to indicate the index in other languages.) The index occurs as the expressioni = 1, which stands for index.
- It is the beginning point for the summation or the lower limit of the summation that is being considered.

This expressionmeans sum the values of x, startingat x 1and ending with x n. |

This expressionmeans sum the values of x, startingat x 1and ending with x 10. |

Thisexpressionmeans sum the values of x, starting at x 3andending with x 10. |

The limits of summation are often understood to mean i =1 through n.Then the notation below and above the summationsign is omitted.Therefore this expression means sum the values ofx, starting at x 1and ending with x n. |

Thisexpression means sum the squared values of x, starting at x 1 and ending with x n. |

Variables inside the summation may be subjected to arithmetic operations of any kind. As an illustration:

This expression means sum the valuesof x, starting at x 1and ending with x nand then squarethe sum. |

It is possible to conduct arithmetic operations on expressions containing more than one variable at the same time. As an illustration:

This expression means form the productof x multiplied by y, starting at x 1and y 1and ending with x nand y nand then sum the products. |

In this expression c is a constant,i.e. an element which does not involve the variable of summation and thesum involves n elements. |

## Problems

Data plus c, which is a constant, equals 11

## 9.2: Summation Notation

The previous part introduced sequences, and the next section will provide the notation and theorems that apply to the sum of terms in a series. Sequences We begin with a definition, which, while scary, is intended to make our life a little bit simpler in the future. Notation for Summarizing When given a sequence (leftright ) and two numbers (m and (p ) that meet the condition (k leq m leq p ), the summation of the sequence from (m to (p) of the sequence (leftright ) is written. The index of summation is the variable denoted by the letter n.

- A short-hand notation for adding up the words of the sequence (left-right ) from a through an is defined by Definition ref, which is merely an abbreviation for the term “ref.” The Greek letter sigma is represented by the symbol (Sigma), which is shorthand for the word’sum’.
- Take, for instance, the sum (a_ +a_ +a_ +a_ +a_ ) written as (a_ = 2n-1) for (ngeq 1), which may be written as ngeq 1.
- As an illustration, Summation notation can be seen in a variety of places, including mathematical definitions.
- The reader is encouraged to make a comparison between this and the information provided in Definition ref.
- Example: The product of the i-th row _(R_ ) of a matrix (A =_) and the (j-th column _(C_ ) of a matrix (B =_) may be written as It is recommended that the reader write down the total and compare it to Definition ref.

Our following example will provide us with some practice with this new notation system. The following is an example of (PageIndex ):

- (displaystyle dfrac)
- (displaystyle dfrac)
- (displaystyle dfrac x-1)n)

- (displaystyle dfrac)
- (displaystyle dfrac)
- (displaystyle dfrac n)

- (1 + 3 + 5 + ldots + 117)
- (1 – dfrac+ dfrac- dfrac+ + – ldots + dfrac)
- (0.09 + 0.09 + 0.009 + ldots 0.09 + 0.009 + ldots 0.09 + 0.009 + ldots 0.09 + 0.09 + 0.009 + ldots 0.09)
- (0.09 + 0.09

- To get to k=4, we insert k=1 into the formula (frac) and add subsequent terms till we get to k=4. Proceeding in the same manner as in (a), we substitute the values (0 through n) for every instance of (n) (4). We recollect the factorials, (n! ), as defined in number Example ref, number refand obtain the following:
- We proceed in the same manner as previously, substituting the index n with the numbers 0 through 5, but emphasizing the variable x, and adding the words arising from this substitution. [=(x-1) – dfrac+ dfrac- dfrac+ dfrac- dfrac+ dfrac- dfrac+ dfrac- dfrac+ dfrac- dfrac+ dfrac- dfrac- dfrac+ dfrac- dfrac- dfrac- dfrac- dfrac- dfrac- dfrac- dfrac- dfrac- dfrac- dfrac- dfrac- Identifying the pattern of the words is essential to write these sums in summation notation correctly. This is accomplished by the effective use of the approaches given in Section ref

beginitem The terms of the sum (1), (3), (5), and so on, form an arithmetic sequence with the initial term a = 1 and the common difference d = 2 as the first and last terms, respectively. Using Equation refto generate a formula for the nth term of the series, we get a formula for the nth term of the sequence: (a = 1 + (n-1)2 = 2n-1) (n geq 1). Now that we know the formula for the terms, which is n-1, and the lower limit of the summation, which is n=1, we can go on to the next step. To complete the issue, we must first find the upper limit of the summation, which we will do in the next step.

When we set (a = 117), we get ((2n-1=117) or ((n = 59)) as the answer.

We rewrite all of the terms as fractions, the subtraction as addition, and the negatives ‘)-)’ with the numerators to obtain the result A geometric sequencefootnote(c = (-1)) may be used to describe the numerators of a number such as (1), (1-1), etc., whereas an arithmetic sequencefootnote(d =n) can be used to describe the denominators of a number such as (1, -1, etc).

- As a result, we obtain the formula (a = frac ) for our terms, and we discover that the lower and higher bounds of summation are, respectively, n=1 and n = 117.
- (n geq 1).
- In order to find the top limit of summing, we must first observe that in order to obtain the n-1 zeros to the right of the decimal point preceding the n, we require a denominator of n times n-1 zeros plus one.
- Because the letter n is used in the bounds of the summation, we must use a different letter for the index of the summation to avoid confusion.
- Several broad characteristics of summation notation are demonstrated in the following theorem: However, while we will not have much use for these qualities in Algebra, they are extremely important in the subject of Calculus.
- We ask the reader to provide evidence to support our findings.
- index

- If a pm b right = a pm b right, then for any real number c, then for any whole number r, then for any real number c, then for any whole number r, then for any real number c, then for any whole number r, then for any real number c, then for any whole number r, then for any real number c, then for any whole number r, then for any real number c, then for any real number

We will now focus our attention to the sums involving arithmetic and geometric sequences in the next section. Let (S) be the sum of the first n terms of an arithmetic sequence (a =a + (k-1) d) for k geq 1; in this case, an is the first term of the series. The letter (S) is written in two distinct ways in order to generate a formula for it. Combining the components that are vertically aligned in these two equations and adding them together, we obtain In this equation, the right-hand side has n terms, all of which are equal to ((2a + (n-1)d)), resulting in the equation (2S = n(2a + (n-1)d).

- The fact that we have represented the total as the product of the number of terms n and the textitof the first and nth terms is a handy approach to remember this last calculation.
- The geometric sum is calculated by starting with a geometric sequence (a = ar).
- After factoring, we get the expression (S(1-r) = an a left(1-r) a right)).
- If we distribute (a) over the numerator, we obtain (a – ar= a_ – a_ ), which provides the formula (a – ar= a_ – a_ ).
- Sums of Arithmetic and Geometric Sequences is an item on the list.

item is the sum (S) of the first n terms of the arithmetic sequence (a_ = a + (k-1)d) for the integer number of terms (k geq 1). If there are n terms in a geometric series (a_ = ar), then the sum of the first n terms is equal to the number of terms in the sequence.

- Item (S = displaystyle a_ = dfrac a =left(dfrac right)) when (r is more than 1), index
- Item (S = displaystyle a_ = sum_ a =n a) when (r is greater than 1), index
- If (r is greater than 1), index
- If (r =1), index
- If (r =1),

We made an honest attempt to derive the formulae in Equation ref, but in order to provide formal proofs, we need the apparatus described in Section ref. Calculus benefits from an application of the arithmetic sum formula, which leads to the discovery of a formula for the sum of the first n natural numbers, which is beneficial in many situations. The natural numbers themselves are a series footnote(1), ldots, ldots, ldots, ldots, ldots, ldots, ldots, ldots, ldots, ldots, ldots, ldots, ldots, ldots, Using Equation ref as a guide, The sum of the first (100) natural numbers, for example, is (frac= 5050).

- It is vital to note that an index textbf is an investment strategy that uses the geometric sum formula as one of its applications.
- Please read the following footnote before proceeding: Consider the following scenario: you have an account with an annual interest rate (r) that is compounded n times per year.
- Consider the following scenario: we want to make regular deposits of (P) dollars at the end of each compounding period.
- Then (A_ = P), since we have made our initial deposit at the beginning of the first compounding period and have not yet earned any interest on our investment.
- On (A_) during the third compounding period, we gain interest, which then rises to (A_ I during the fourth compounding period.
- This trend continues until the conclusion of the (k)th compounding, at which point we have The sum in the parentheses above represents the sum of the first k terms of a geometric sequence with (a = 1) and (r = frac) as the first and second terms, respectively.
- The Present Value of an Ordinary Annuity in the Future Consider the case of an annuity that pays an annual interest rate (r) that is compounded n times per year.
- In the case of periodic deposits (P) made at the conclusion of each compounding period, the amount (A) remaining in the account after t years is given by the expression In order to simplify Equation ref, the reader is invited to put I = frac) into the equation and simplify.
- Last but not least, if the deposit (P) is placed at the beginning of the compounding period rather than at the end, the annuity is referred to as an indextextbf.

For the sake of the reader, we will not go into detail about how we came up with the formula for calculating the future value of an annuity-due. As an illustration (PageIndex ) An typical annuity pays a compounded monthly interest rate of (6% per year) on the principal invested.

- If monthly payments of ()50) are paid, calculate the value of the annuity in (30) years
- How many years will it take for the annuity to grow to a value of ()100,000
- And how many years will it take for the annuity to grow to a value of ()100,000

- We have (r = 0.06) and (n = 12), which means that I = frac= frac= 0.005
- R = 0.06
- N = 12) With (P=50) and (t=30), the following is true: We have arrived at the final solution of ()50,!225.75)
- Setting (A = 100000) and solving for (A = 100000) will give us the answer to how long it will take for the annuity to grow to ((100,000)). (t). It is necessary to isolate the exponential and calculate the natural logs on both sides of the equation.

The investment will rise to a total value of ()100,000 in slightly over (40) years, according to this calculation. When we compare this to our response to part 1, we can see that the value of the annuity practically doubles in just ten extra years. This is a lesson that should be remembered. We will conclude this part with a little glimpse into Calculus by looking into textitsums, also known as indextextbf. Consider the value of the number (0.overline ). This number can be represented as We know from Example ref that we can express the sum of the first n of these terms as the following: Using Equation ref, we obtain the following: Given that (0.overline ) is the same value as (1-frac), it follows that (0.overline is the same value as (n rightarrow infty).

- Some readers may find it distressing that we have just argued that (0.overline=1), which we apologize for.
- In this part, we will discuss a geometric series theorem that will be useful in the future.
- Suppose you have the series (a = ar) for (k geq 1), where (|r|1), and you want to get the sum of a, ar, ar2, and dots.
- To justify the conclusion in Theorem ref, we use the formula in Equation reffor the sum of the first (n) terms of a geometric sequence, and we examine the formula as if it were a geometric sequence with n initial terms (n rightarrow infty).
- (n rightarrow infty).
- geq 1), although we’ll look at a few examples in the exercises.

## Contributors and Attributions

- The investment will increase to a total value of ()100,000 in slightly over (40) years, assuming no inflation. We can see that the value of the annuity practically doubles in just ten extra years when we compare this to our response to Part 1. Remembering this is a valuable life lesson. In order to conclude this lesson, we will take a brief detour into Calculus by looking into textitsums, also known as indextextbf. Take the number for example (0.overline ). This number can be written as Based on Example ref, we know that the sum of the first n of these phrases may be written as We have the following using Equation ref: Given that (0.overline ) is the same value as (1-frac), it follows that (0.overline is the same value as frac (n rightarrow infty). Section refteaches us that if (n – fracrightarrow 1 ) equals (n – fracrightarrow 0), then ((1 – fracrightarrow 0) equals (nrightarrow infty) equals (n – fracrightarrow 0). Some readers may find it distressing that we have just argued that (0.overline=1), but that is not our intention. For the sake of convenience, it is often assumed that (0.overline= frac) so that (0.overline= 3left(0.overline right) = 3left(0.overline frac) = 1)
- (0.overline= 3left(0.overline frac) = 1)
- And (0.overline= 3left(0.overline frac) = 1)
- (0 It is possible to think of every non-terminating decimal as being composed of an infinite sum whose denominators are powers of (10)
- As a result, the phenomena of adding up infinitely many terms and arriving at a finite number is not as strange as it may look. In this part, we will discuss a geometric series theorem that will conclude the chapter. Definition: Geometric Series is a collection of geometric figures. Suppose you have the series (a = ar) for (k geq 1), where (|r|1), and you want to get the sum of a, ar, ar2, and dots. If you have the sequence a, ar, and dots, you want to find the total of (a + ar + ar2 + dots. To justify the conclusion in Theorem ref, we use the formula in Equation reffor the sum of the first (n) terms of a geometric sequence, and we examine the formula as if it were a geometric sequence with (n) terms (n rightarrow infty). Based on the assumption that r1 equals -1r1, we may interpret 0 as r rightarrow (n rightarrow infty). This is denoted by the notation (n “rightarrow “) infty. For the most part, we’ll leave it to Calculus to figure out what went wrong when (|r| geq 1), although we’ll look into it a little bit in the exercises.

## Sigma notation for sums – Topics in precalculus

22Three theoremsThe sum of consecutive numbersRemainder classes modulomAn arithmetic seriesT HIS—Σ—is the Greek letter sigma. We use it to indicate a sum.For example:This means that we are to repeatedly addkak.The first time we write it, we putk= 1.That is indicated by thelower indexof the letter sigma. The next time, we putk= 2, then 3, and so on, until we come to theupper index, which in this case is4.In other words, we are to repeatedly addkak, whichwe call theargumentof the sum, or thesummand, starting withk= 1 and ending withk= 4.kis called adummy indexbecause it does not actually affect the sum, and we could indicate that sum using any letter we please; for examplej:Problem 1.Write the following sums.To see the answer, pass your mouse over the colored area.To cover the answer again, click “Refresh” (“Reload”). Do the problem yourself first! |

a) | 2 ·1+2 ·2+2 ·3+2 ·4+2 ·5 | |

= | 2 + 4 + 6 + 8 + 10. |

In this example, the upper indexn is used. That is how you would represent the sum of successive whole integers in sigma notation. Isn’t the final term of the total interesting?

We designate the next-to-last position as (n 1). Problem 2.Compute the following sum in your own words. The sum begins with j=0 in this case. That is how we represent a polynomial in degreen using the sigma notation (in degrees). The sixth problem in Topic 6 is:

When the signals alternate in this manner, both positively and negatively, we refer to this as an alternating series. (1)k is the factor that causes the signals to alternate. When ki is zero or an even integer, then (1)k will equal +1 in this case. However, if the number is odd, (1)kwill be negative. See Lesson 13 of Algebra, Example 1, and Problem 7 for more information. Problem 3.Compute the following amount in your own words. In this example, we’ll use the sigma notation to represent the total of two plus four plus six plus eight plus.

- An even number, which is a multiple of 2, is denoted by the numeral 2.
- For the higher index, we can determine that it must be 50 because we must have 2 k= 100 in order for the equation to work.
- Problem #4: To represent these sums, use the sigma notation.
- A sum that is made up of words may be divided into a sum over each term and a sum over the entire amount.

= | a1+b1+a2+b2+. +an+bn | |

= | a1+a2+. +an+b1+b2+. +bn | |

= |

If the argument of the summation is a constant, that is, if it does not change depending on the indexk, then the sum is equal to the number of terms multiplied by the constant in question. Numbers added together in a series We learned in the Appendix to Arithmetic that the sum of successive integers – an atriangular number – may be calculated using the following formula: 12 n (n+ 1), where n is the number that comes last in the total. To put it another way, Exhibit 4 is a good example. Cite the following summation theorems in order to demonstrate:

Proof. | = | Theorems 1 and 2 | ||

= | 6 ·½ n (n+ 1) − 5 n. | ThesumandTheorem 3 | ||

= | 3 n2+ 3 n− 5 n | |||

= | 3 n2− 2 n | |||

= | n (3 n− 2). |

Modulom is a subclass of the remaining classes. When a number is divided by three, for example, the remaining will be one of the numbers 0, 1, or 2. Here is the remainder class, with the remainder value of 0:369121518. According to algebra, these are the numbers 3 k.k= 1, 2, 3,. and so on. We refer to these as the integers that are congruent to 0modulo 3. Here is the remaining class with the remaining fraction of 1:1471013. Alternatively, starting withk=1, these are the numbers 3 k+1 – alternatively, to begin withk=2, these are the numbers 3 k-2.

- These are the integers that are congruent to the modulo 3 of 1.
- Finally.
- Listed below are the integers that are congruent to 2(orto 1).
- Those are the three remaining classesmodulo 3, that is, after the division by three has been performed.
- 0, 1, 2, 3, 4 are the digits.

51015201611162712173813 184914 51015201611162712173813 51015201611162712173813 51015201611162712173813 51015201611162712173813 51015201611162712173813 51015201611162712173813 51015201611162712173813 51015201611162712173813 51015201611162712173813 19.5c)Indicate each remainder class algebraically, and let each one beginc)with k= 1.5 k5 k45 k35 k25 k 19.5c)Indicate each remainder class algebraically, and let each one beginc)with Series of mathematical operations An anarithmetic series is a sum in which each term is formed from the preceding term by adding the same number to the end of the series.

- The constant difference is the number that represents this difference.
- + 32 is a prime number.
- The constant difference is represented by the number 3.
- Example 5.Prove that 2 + 5 + 8 + 11 + 14 +.
- 2 + 5 + 8 + 11 + 14 +.
- Solution.
- They have the algebraic form3 k 1, which means that they are equal.

As a result, we must demonstrate: Here’s some evidence to back up my claim: Demonstrate that every square number is the sum of successive odd integers in problem 6. That example, the sum of 1 + 3 + 5 +. + (2 n 1) =n2 is equal to n2. Proof:

In Topic 27, Mathematical Induction, the third problem is referred to as Problem 3. And in the Appendix of Arithmetic, we provide a precise explanation of why this is the case. Problem number seven. Prove: 1 + 5 + 9 +. + (4 n 3) =n (2 n 1). 1 + 5 + 9 +. + (4 n 3) =n (2 n 1). Proof:

= | 4 ·½ n (n+ 1) − 3 n | |

= | 2 n2+ 2 n− 3 n | |

= | 2 n2−n | |

= | n (2 n− 1). |

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Copyright © 2021Lawrence SpectorQuestions or comments?E-mail:[email protected]

## Series and Their Notations

- Make use of summation notation
- Calculate the sum of the firstnterms in an arithmetic series using the summation formula
- In a geometric series, you may calculate its total by using the sum of the firstnterms formula
- Use the formula for the sum of an infinite geometric series to solve your problem. Identify and resolve annuity issues.

For their daughter’s future college expenses, a couple chooses to form a college fund. Every month, they intend to contribute $50 to the fund. The fund pays an annual interest rate of 6 percent, which is compounded monthly. What amount of money will they have saved by the time their daughter is ready to start college in six years is anyone’s guess. Throughout this section, we will learn how to respond to this question effectively. Taking into account the quantity of money invested as well as interest earned is necessary in order to accomplish this goal.

## Arithmetic Series

In order to calculate the entire amount of money in the college fund and the sum of the amounts deposited, we must add the amounts deposited each month and the amounts earned each month to each other. Aseries is a word that refers to the sum of the terms in a sequence. Take, for example, the following succession of sentences. 3+7+11+15+19+cdots The partial sum of a series is the sum of a finite number of successive terms that begin with the first term in the series. The notation is as follows: begin text =3 =3+7=10 =3+7+11=21 =3+7+11+15=36 end text =3 =3+7=10 =3+7=21 =3+7+15=36 When representing a series, the summation notation is employed.

- It comprises an explicit formula that specifies the beginning and last terms in the series, as well as the order in which they appear.
- A variable referred to as the index of summation is written underneath the sigma symbol.
- The number that is greater than the sigma, known as the upper limit of summation, is the number that is utilized to construct the last term in a sequence of terms.
- Beginning with inserting the terms fork and listing the terms of this series, we may go on to the next step.
- 2k=2+4+6+8+10=30

### A General Note: Summation Notation

When calculating the total amount of money in the college fund, it is necessary to put together the amounts deposited each month as well as the amounts earned in each month. Aseries is a word that refers to the total of the terms of a sequence. To illustrate, consider the following set of sentences. 3+7+11+15+19+cdots Textpartial sum of a series is the sum of a finite number of consecutive terms that begin with the first term in the series. The notation is as follows: begin text =3 =3+7=10 =3+7+11=21 =3+7+11+15=36 end text =3 =3+7=10 =3+7+11=36 Series are represented using summation notation.

- When a sequence of terms is written in summation notation, the beginning and end terms in the series are specified.
- Underneath the sigma, a variable called the index of summation is written.
- It is this number that is employed to construct the final term in a series, which is referred to as the upper limit of summation.
- Begin with =2 left(1 right)=2, continue with =2 left(2 right)=4, continue with =2 left(3 right)=6, continue with =2 left(4 right), and conclude with =2 left(5 right).

By combining the following terms, we can obtain the sum of the series. Limits on the total amount of money that can be spent on one item. 2k=2+4+6+8+10=30

### QA

No. The lower limit of the summing can be any integer, although 1 is a common choice because of its simplicity. As an alternative to the limit of one, we shall consider cases with lower limits of summation other than one.

### How To: Given summation notation for a series, evaluate the value.

- Determine the lower limit of the summing process
- Determine the highest limit of the summing process
- Each value ofk between the lower and higher limits should be substituted into the formula as follows: To obtain the total, add the numbers together.

### Example: Using Summation Notation

Calculate the sum of the limits .

### Try It

Calculate the total of the limits

### Arithmetic Series

In the same way that we looked at different sorts of sequences, we will look at different forms of series. Remember that anarithmetic sequence is a series in which the difference between any two successive terms is equal to the common difference,d, and thus The sum of the terms of an arithmetic sequence is referred to as an anarithmetic series in mathematical jargon. It is possible to represent the sum of the firstnterms of an arithmetic series as: = +left( +dright)+left( +2dright)+.+left( -dright)+ +left( -dright).

The sum of the firstnterms of an arithmetic series may be calculated by adding these two expressions for the sum of the firstnterms of an arithmetic series together.

begin = +left( +dright)+left( +2dright)+.+left( -dright)+ + = +left( -dright)+.+left( +2dright)+.+left( -dright)+ + = +left( -dright)+left( -2dright)+.+left( +dright)+ + _hline _hline _hline _hline _hline _hline _hline _hline _hline _hline 2 =left( + right) 2 =right( + right) 2 =left( + right) 2 =left( + right) 2 =left( + right) 2 =left( + right) 2 =left( + right) 2 =left( + right) 2 +left( + right)+.+left( + right)+.+left( + right)+.+left( + right).+left( + right).+left( + right).+left( + right).+left( + right).+left( + For this total, we can reduce it to 2 =nleft( + 2 right) because there arenterms in the series.

To determine the formula for the sum of the firstnterms of an arithmetic series, we divide the number by two.

### A General Note: Formula for the Partial Sum of an Arithmetic Series

The sum of the terms of an arithmetic sequence is known as an anarithmetic series. The partial sum of an arithmetic sequence can be calculated using the formula =dfrac + right)

### How To: Given terms of an arithmetic series, find the partial sum

- Figure out what you need to know and how to get it. Then, substitute the appropriate values for , andninto the formula =dfrac + (right)
- Make it easier to find_

### Example: Finding the partial sum of an Arithmetic Series

Find the partial sum of each arithmetic series in the given time frame.

- 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32
- 20 + 15 + 10 + dots + -50
- 3k – 8
- 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32
- 5 + 8 + 11 + 14 + 17 + 20 +

### Try It

To determine the partial sum of each arithmetic series, use the formula provided. 1.4+1.6+1.8+2.0+2.2+2.4+2.6+2.8+3.0+3.2+3.412+21+29dots + 69sumlimits 5 – 6k 1.4+1.6+1.8+2.0+2.2+2.4+2.6+2.8+3.0+3.2+3.412+21+29dots + 69sumlimits 5 – 6k

### Example: Solving Application Problems with Arithmetic Series

A lady is able to walk a half-mile on Sunday after having minor surgery, which she does on Saturday. Every Sunday, she adds an additional quarter-mile to her daily stroll. What do you think the total number of kilometers she has walked will be after 8 weeks?

### Try It

In the first week of June, a man receives $100 in pay. The amount he makes each week is $12.50 greater than the previous week. How much money has he made after 12 weeks of work?

## Geometric Series

In the same way that the sum of the terms in an arithmetic sequence is referred to as an arithmetic series, the sum of the terms in a geometric sequence is referred to as an ageometric series (or ageometric sequence). It’s important to remember that an ageometric sequence is a series in which the ratio of any two successive terms is equal to the common ratio, r, and thus As an example, we may express the sum of the firstnterms in a geometric series as = + +.+ . It is possible to use algebraic manipulation to develop a formula for the sum of the firstnterms of a geometric series in the same way that we did with arithmetic series.

- r = r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r+ r Then we subtract this equation from the original equation to get the result.
- +.
- +.
- +.
- +.

To derive a formula for_, factora 1on the right hand side and divide both sides by the left hand side of the equation (1-rright). >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac >=dfrac

### A General Note: Formula for the Sum of the FirstnTerms of a Geometric Series

An ageometric series is a mathematical word that refers to the sum of the terms in a geometric sequence, in the same way that an arithmetic series refers to the total of the terms in arithmetic sequence. It’s important to remember that an ageometric sequence is a sequence in which the ratio of any two successive words is equal to the common ratio, orr. The sum of the firstnterms of a geometric series can be written as = + r+ +.+ + . We can use algebraic manipulation to obtain a formula for the sum of the firstnterms of a geometric series in the same way that we do with arithmetic series.

the value of the variable is r = the value of the variable is r plus the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of the variable is the value of Afterwards, we subtract this equation from the original equation to arrive at the final equation.

begin ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• The following is an example of how to use the symbol for “right”: (r” + “+” +.+ “r”).

When obtaining a formula for_, factora 1on the right hand side and divide both sides by the number on the left (1-rright).

### How To: Given a geometric series, find the sum of the firstnterms.

- Identify ,r,text n
- Identify ,r,text n
- Fill in the blanks with the values for ,r, andninto the formula =dfrac left(1- dfrac right)
- Make it easier to find_

### Example: Finding the FirstnTerms of a Geometric Series

To obtain the indicated partial sum of each geometric series, use the formula to the right.

- Calculate the partial sum of each geometric series using the formula provided.

### Try It

Utilize the formula to determine the suggested partial sum of each geometric series. for the series1 text 000 + 500 + 250 + dots use the formula to determine the indicated partial sum of each geometric series. In order to find the sumsumlimits, the following formula should be used:

### Example: Solving an Application Problem with a Geometric Series

An employee’s beginning wage is $26,750 when they start a new job. Every year, he earns a 1.6 percent pay boost. Calculate his total earnings over the course of five years.

### Try It

A new employee’s beginning wage is $32,100 upon commencing a new position.

Every year, she earns a 2 percent pay boost. How much money will she have made at the conclusion of the eight-year period?

## Using the Formula for the Sum of an Infinite Geometric Series

Until now, we’ve just looked at finite series as a starting point. The total of all the terms in an infinite series, as opposed to just the sum of the firstnterms, is something we are interested in in some situations. An infinite series is the sum of all the terms in an infinite sequence, which is a mathematical construct. The numbers 2+4+6+8+dots are an example of an infinite series. This series can alternatively be expressed in summation notation as sumlimits 2k, where the upper limit of summation is infinity, and where the lower limit of summation is infinite.

Therefore, the whole value of this infinite series is not known at this time.

### Determining Whether the Sum of an Infinite Geometric Series is Defined

When the terms of an infinite geometric series approach zero, it is possible to define the sum of an infinite geometric series. Each of the entries in this series approaches the value of 0:1 + 0.2 + 0.04 + 0.008 + 0.0016 + dots The common ratio is equal to 0.2. When Asngets big, the values of ofrnget extremely tiny and close to zero, asngets. Each consecutive term has a less impact on the total than the one before it. In order to go closer to zero with each subsequent term, the total of the terms approaches a finite value.

### DETERMINING WHETHER THE SUM OF AN INFINITE GEOMETRIC SERIES IS DEFINED

If the series is geometric and-1 r 1, the total of an infinite series can be defined as -1 r 1.

### How To:Given the first several terms of an infinite series, determine if the sum of the series exists.

- Calculate the relationship between the second term and the first term. Calculate the relationship between the third term and the second term. Continue in this manner to guarantee that the ratio of a term to the phrase before it remains consistent throughout the procedure. Assuming that a common ratio,r, was discovered in step 3, check to determine if-1 r 1 is true
- Otherwise, the series is geometric. If this is the case, the total is defined. If this is not the case, the total is not specified.

### Example: Determining Whether the Sum of an Infinite Series is Defined

Check to see if the sum of each infinite series has a definite value or not.

- 12+8+4+/dots
- Dfrac+dfrac+dfrac+dots
- Dfrac+dfrac+dfrac+d

### try it

Check to see if the sum of the infinite series has a specified definition.

- Dfrac+dfrac+dfrac+dfrac+cdots
- 24+(-12)+6+(-3)+cdots
- Dfrac+dfrac+dfrac+cdots
- Dfrac+dfrac+dfrac+cd

## Finding Sums of Infinite Series

When the sum of an infinite geometric series is known, we may compute the sum of the infinite geometric series. There is a relationship between the sum of an infinite series and the sum of firstnterms of a geometric series in the formula for the sum of an infinite series. Left(1- right) =dfrac left(1- right). We will look at an infinite series withr=frac as an example. What happens when the amount of tornasnincreases? beginright)= fracright)= fracright)= fracendright)= fracright)= fracendright)= fracright)= fracendright)= fracendright)= fracendright)= fracendright)= fracendright)= fracendright)= fracendright)= fracendright)= fracendright)= fracendright)= fracendright)= The value ofrndecreases at an alarming rate.

begin(right)= frac024; right)= frac048; text 576; right)= frac073; text 741; text 824; end(right)= frac024; frac048; text 576; frac073; text 741; text 824; frac024; text 576; frac048; text 576; frac024; text 576; frac048; text Asngets are extremely huge, whereas rngets are quite little.

Asrnapproaches 0,1-rnapproaches 1, asrnapproaches 0,1-rnapproaches 1. When this occurs, the numerator approaches the value of a 1. This provides us the formula for the sum of an infinite geometric series, which we can use in our calculations.

### A General Note: FORMULA FOR THE SUM OF AN INFINITE GEOMETRIC SERIES

The following is the formula for the sum of an infinite geometric series with-1 r 1: S=dfrac

### How To:Given an infinite geometric series, find its sum.

- Make note of the numbers fora 1andr
- Confirm that-1 r 1
- Substitute them into the formula S=dfrac
- And finally, check your work. Make it easier to findS

### Example: Finding the Sum of an Infinite Geometric Series

Find the sum of the following numbers, if one exists:

- 10+9+8+7+dots
- 248.6+99.44+39.776+dots
- Sumlimits 4text 374cdotleft(-dfracright)
- Sumlimits dfraccdotleft(-dfracright)
- Sumlimits dfraccdot

### Example: Finding an Equivalent Fraction for a Repeating Decimal

Find the fraction that is comparable to the repeating decimal0.overline

### try it

Determine whether or not the sum exists.

- 2+dfrac+dfrac+dots
- Sumlimits
- Sumlimits left(-dfracright)k
- Sumlimit

Solution/reveal-answer]

## Annuities

A issue in which a couple put a certain amount of money each month into a college fund for a period of six years was discussed at the outset of the section. An annuity is a type of investment in which the purchaser makes a series of periodic, equal payments over a period of time. In order to calculate the value of an annuity, we must first calculate the sum of all the payments made as well as the interest generated. In this scenario, the couple makes a monthly investment of $50. The amount of the original deposit is represented by this figure.

- In order to calculate the interest rate per payment period, we must divide the annual percentage interest (APR) rate of 6 percent by the number of payment periods.
- We may calculate the worth of the account after interest has been applied by multiplying the amount in the account each month by 100.5 percent.
- Let’s see if we can figure out how much money is in the college fund and how much interest has been earned.
- During six years, there are 72 months, which equals 72 sons.
- =dfrac right)approx 4 text 320.44 =dfrac 320.44 =dfrac 320.44 Text 320.44 =dfrac 320.44 =dfrac 320.44 Text 320.44 =dfrac 320.44 Text 320.44 =dfrac 320.44 Text 320.44 After making the last transfer, the couple will have a total of $4,320.44 in their bank account after expenses.
- Thus, as a result of the annuity, the couple earned $720.44 in interest on their college savings account, which they invested.

### How To: Given an initial deposit and an interest rate, find the value of an annuity.

- A issue in which a couple put a certain amount of money each month into a college fund for a period of six years was discussed at the outset of this section. Purchasing an annuity entails making a series of periodic, equal payments over a period of time. If we want to figure out the value of an annuity, we must add up all the payments and interest generated and then multiply that total by 100. Monthly investments of $50 are shown in this example. The amount of the first deposit is represented by this figure: In addition to the monthly compounded income, the account earned 6 percent interest every year. In order to calculate the interest rate per payment period, we must divide the annual percentage interest (APR) rate of 6 percent by the number of payment periods in a month. As a result, the interest rate is 0.5 percent each month on average. It is possible to calculate the worth of the account once interest has been applied by multiplying each month’s deposit amount by 100.5 percent. Using a geometric series with =50andr=100.5 percent =1.005, we can calculate the amount of the annuity immediately following the final deposit. The value of the annuity will be $50 following the initial deposit. Look into determining the amount of money in the college fund and how much interest has been produced on the money. Using the formula for the sum of the firstnterms of a geometric series, we can determine the value of the annuity afterndeposits.n There are 72 months in six years, thus son=72. It is possible to simplify the formula by substituting =50, r=1.005, and n=72 into it, and the amount of the annuity after 6 years will be determined. =dfrac right)approx 4 text 320.44 =dfrac 320.44 =dfrac 320.44 Text 320.44 =dfrac 320.44 =dfrac 320.44 =dfrac 320.44 =dfrac 320.44 text 320.44 text 320.44 text 320.44 text 320.44 text In all, the pair will have $4,320.44 in their account after making the last contribution. Take note that the couple paid 72 installments of $50 apiece, for a total of 72left(50right) = $3,600 in payments. Essentially this implies that the pair received $720.44 in interest on their education fund as a result of the annuity they purchased.

- Subtract the yearly interest rate from the number of times per year that interest is compounded to arrive at the answer. In the case of findr, add 1 to this amount.

Calculate a geometric series by substituting values for , r, and n into the formula for the sum of the firstn terms of a geometric series, which is =dfrac left(1- dfrac right). Calculate the value of the annuity afterndeposits in the simplest possible way_.

### Example: Solving an Annuity Problem

Every month at the beginning of the month for the next ten years, a $100 deposit is made into a college fund.

A 9 percent annual interest rate is earned on the fund, which is compounded monthly and paid at the end of each month. What amount of money is currently in the account following the most recent deposit?

### Try It

Every month, $200 is put into a retirement account at the beginning of the month. Each month, the fund receives 6 percent annual interest, which is compounded and deposited into the account at the end of each month. What amount of money will be in the account after ten years of deposits?

## Key Equations

sum of the firstn terms of an arithmetic series | _ =dfrac _ + _ right)} |

sum of the firstn terms of a geometric series | _ =dfrac _ left(1- ^ right)}, rne 1 |

sum of an infinite geometric series with-1 r 1 | _ =dfrac _ } |

## Key Concepts

- A series is defined as the total of all of the words in a sequence. Summation notation is a standard notation for series that employs the Greek letter sigma to indicate the sum of the numbers in the series. An arithmetic series is the sum of the terms in an arithmetic sequence
- It is also known as arithmetic sequence sum. A formula may be used to find the sum of the firstnterms of an arithmetic series
- For example, A geometric series is the sum of the terms in a geometric sequence
- It is also known as a geometric sequence. A formula may be used to get the sum of the firstnterms of a geometric series
- For example, The sum of an infinite series occurs if the series is geometric with-1 r 1
- Otherwise, the sum does not exist. Using a formula, one may determine whether or not the sum of an infinite series exists. An annuity is a type of investment account into which the investor makes a series of payments on a regular basis. The value of an annuity may be calculated with the use of geometric series.

## Glossary

Investment in which the purchaser makes a series of periodic, equal payments is referred to as an annuity. set of mathematical operations when the terms of an arithmetic series are added together diverge When the total of a series does not equal a real number, the series is said to diverge. sequence of geometric figures when the terms of a geometric series are added together The summation index is an index of the summation. Summation notation refers to a variable that is employed in an explicit formula for the terms of a series and written below the sigma, with the lower limit of summationinfinite series as the lower limit of the summation infinity series when there are an unlimited number of terms in a series upper and lower limits of the summation This is the number that is used in the explicit formula to determine the first term in a sequence of terms.

nth partial sum (nth partial sum) an expression expressing the sum of the firstnterms of a sequenceseries the total of all the words in a succession of phrases summation notation is a type of notation that summarizes anything.

the maximum amount of summation that may be made This is the number that is used in the explicit formula to get the final term in a series of terms.