Sequences, Probability, and Counting Theory

# Series and Their Notations

### Learning Objectives

• Use summation notation.
• Use the formula for the sum of the ﬁrst n terms of an arithmetic series.
• Use the formula for the sum of the ﬁrst n terms of a geometric series.
• Use the formula for the sum of an inﬁnite geometric series.
• Solve annuity problems.

### Using the Formula for Geometric Series

Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio, We can write the sum of the first terms of a geometric series as

Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the firstterms of a geometric series. We will begin by multiplying both sides of the equation by

Next, we subtract this equation from the original equation.

Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for divide both sides by

### Formula for the Sum of the First n Terms of a Geometric Series

A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the firstterms of a geometric sequence is represented as

### How To

Given a geometric series, find the sum of the first n terms.

1. Identify
2. Substitute values for and into the formula
3. Simplify to find

### Finding the First n Terms of a Geometric Series

Use the formula to find the indicated partial sum of each geometric series.

1. for the series

1. and we are given that

We can find by dividing the second term of the series by the first.

Substitute values for into the formula and simplify.

2. Find by substituting into the given explicit formula.

We can see from the given explicit formula that The upper limit of summation is 6, so

Substitute values for and into the formula, and simplify.

Use the formula to find the indicated partial sum of each geometric series.

for the series

9,840

### Solving an Application Problem with a Geometric Series

At a new job, an employee’s starting salary is $26,750. He receives a 1.6% annual raise. Find his total earnings at the end of 5 years. [reveal-answer q=”fs-id1165137637577″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137637577″] The problem can be represented by a geometric series with $n=5\text{;}\,$and Substitute values for$r\text{,}$ and into the formula and simplify to find the total amount earned at the end of 5 years. He will have earned a total of$138,099.03 by the end of 5 years.[/hidden-answer]

### Try It

At a new job, an employee’s starting salary is $32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years? [reveal-answer q=”fs-id1165135526997″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135526997″]$275,513.31

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

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is

This series can also be written in summation notation as where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.

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

If the terms of an infinite geometric series approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:

The common ratio
As gets very large, the values of get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with approach 0; the sum of a geometric series is defined when

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

The sum of an infinite series is defined if the series is geometric and

### How To

Given the first several terms of an infinite series, determine if the sum of the series exists.

1. Find the ratio of the second term to the first term.
2. Find the ratio of the third term to the second term.
3. Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.
4. If a common ratio, was found in step 3, check to see if . If so, the sum is defined. If not, the sum is not defined.

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

Determine whether the sum of each infinite series is defined.

1. The ratio of the second term to the first is
which is not the same as the ratio of the third term to the second, The series is not geometric.
2. The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of The sum of the infinite series is defined.
3. The given formula is exponential with a base of the series is geometric with a common ratio of The sum of the infinite series is defined.
4. The given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot yield a finite sum.

Determine whether the sum of the infinite series is defined.

### Try It

The sum is not defined.

### Try It

The sum of the infinite series is defined.

### Try It

The sum of the infinite series is defined.

#### Finding Sums of Infinite Series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first terms of a geometric series.

We will examine an infinite series with What happens to as increases?

The value ofdecreases rapidly. What happens for greater values of

As gets very large, gets very small. We say that, as increases without bound, approaches 0. As approaches 0, approaches 1. When this happens, the numerator approaches This give us a formula for the sum of an infinite geometric series.

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

The formula for the sum of an infinite geometric series with is

### How To

Given an infinite geometric series, find its sum.

1. Identifyand
2. Confirm that
3. Substitute values for and into the formula,
4. Simplify to find

### Finding the Sum of an Infinite Geometric Series

Find the sum, if it exists, for the following:

1. There is not a constant ratio; the series is not geometric.
2. There is a constant ratio; the series is geometric. and so the sum exists. Substitute and into the formula and simplify to find the sum:

3. The formula is exponential, so the series is geometric with Find by substituting into the given explicit formula:

Substitute and into the formula, and simplify to find the sum:

4. The formula is exponential, so the series is geometric, butThe sum does not exist.[/hidden-answer]

### Finding an Equivalent Fraction for a Repeating Decimal

Find an equivalent fraction for the repeating decimal

We notice the repeating decimal so we can rewrite the repeating decimal as a sum of terms.

Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0.1 in the second term, and the second term multiplied to 0.1 in the third term.

Notice the pattern; we multiply each consecutive term by a common ratio of 0.1 starting with the first term of 0.3. So, substituting into our formula for an infinite geometric sum, we have

Find the sum, if it exists.

3

### Try It

The series is not geometric.

### Solving Annuity Problems

At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example, the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added. We can find the value of the annuity right after the last deposit by using a geometric series with and After the first deposit, the value of the annuity will be$50. Let us see if we can determine the amount in the college fund and the interest earned.

We can find the value of the annuity after deposits using the formula for the sum of the first terms of a geometric series. In 6 years, there are 72 months, so We can substitute into the formula, and simplify to find the value of the annuity after 6 years.

After the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of$50 each for a total of 72(50) = $3,600. This means that because of the annuity, the couple earned$720.44 interest in their college fund.

### How To

Given an initial deposit and an interest rate, find the value of an annuity.

1. Determinethe value of the initial deposit.
2. Determinethe number of deposits.
3. Determine
1. Divide the annual interest rate by the number of times per year that interest is compounded.
2. Add 1 to this amount to find
4. Substitute values for
into the formula for the sum of the first terms of a geometric series,
5. Simplify to find the value of the annuity after deposits.

### Solving an Annuity Problem

A deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit? [reveal-answer q=”fs-id1165135388500″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135388500″] The value of the initial deposit is$100, soA total of 120 monthly deposits are made in the 10 years, so To find divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit.

Substituteinto the formula for the sum of the first terms of a geometric series, and simplify to find the value of the annuity.

So the account has $19,351.43 after the last deposit is made.[/hidden-answer] ### Try It At the beginning of each month,$200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years?

$32,775.87 [/hidden-answer] Access these online resources for additional instruction and practice with series. ### Key Equations  sum of the first terms of an arithmetic series sum of the first terms of a geometric series sum of an infinite geometric series with ### Key Concepts • The sum of the terms in a sequence is called a series. • A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See (Figure). • The sum of the terms in an arithmetic sequence is called an arithmetic series. • The sum of the firstterms of an arithmetic series can be found using a formula. See (Figure) and (Figure). • The sum of the terms in a geometric sequence is called a geometric series. • The sum of the firstterms of a geometric series can be found using a formula. See (Figure) and (Figure). • The sum of an infinite series exists if the series is geometric with • If the sum of an infinite series exists, it can be found using a formula. See (Figure), (Figure), and (Figure). • An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See (Figure). ### Section Exercises #### Verbal What is an partial sum? [reveal-answer q=”fs-id1165134387624″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165134387624″] An partial sum is the sum of the first terms of a sequence. [/hidden-answer] What is the difference between an arithmetic sequence and an arithmetic series? What is a geometric series? [reveal-answer q=”fs-id1165135435790″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135435790″] A geometric series is the sum of the terms in a geometric sequence. [/hidden-answer] How is finding the sum of an infinite geometric series different from finding the partial sum? What is an annuity? [reveal-answer q=”fs-id1165135487097″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135487097″] An annuity is a series of regular equal payments that earn a constant compounded interest. [/hidden-answer] #### Algebraic For the following exercises, express each description of a sum using summation notation. The sum of terms from to The sum from of to of [reveal-answer q=”fs-id1165135205629″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135205629″] [/hidden-answer] The sum of from to The sum that results from adding the number 4 five times [reveal-answer q=”fs-id1165135333040″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135333040″] [/hidden-answer] For the following exercises, express each arithmetic sum using summation notation. [reveal-answer q=”fs-id1165137696462″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137696462″] [/hidden-answer] For the following exercises, use the formula for the sum of the first terms of each arithmetic sequence. [reveal-answer q=”fs-id1165137871515″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137871515″] [/hidden-answer] [reveal-answer q=”fs-id1165137619905″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137619905″] [/hidden-answer] For the following exercises, express each geometric sum using summation notation. [reveal-answer q=”fs-id1165137456659″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137456659″] [/hidden-answer] For the following exercises, use the formula for the sum of the first terms of each geometric sequence, and then state the indicated sum. [reveal-answer q=”fs-id1165135338212″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135338212″] [/hidden-answer] [reveal-answer q=”fs-id1165137704468″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137704468″] [/hidden-answer] For the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not, state the reason. [reveal-answer q=”fs-id1165135496306″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135496306″] The series is defined. [/hidden-answer] [reveal-answer q=”fs-id1165137701119″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137701119″] The series is defined. [/hidden-answer] #### Graphical For the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened the account with an initial deposit of$50. Each month thereafter he increased the previous deposit amount by $20. Graph the arithmetic sequence showing one year of Javier’s deposits. Graph the arithmetic series showing the monthly sums of one year of Javier’s deposits. [reveal-answer q=”fs-id1165137828079″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137828079″][/hidden-answer] For the following exercises, use the geometric series Graph the first 7 partial sums of the series. What number does seem to be approaching in the graph? Find the sum to explain why this makes sense. [reveal-answer q=”fs-id1165135208860″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135208860″] Sample answer: The graph of seems to be approaching 1. This makes sense becauseis a defined infinite geometric series with [/hidden-answer] #### Numeric For the following exercises, find the indicated sum. [reveal-answer q=”fs-id1165137453619″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137453619″] 49 [/hidden-answer] [reveal-answer q=”fs-id1165134183115″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165134183115″] 254 [/hidden-answer] For the following exercises, use the formula for the sum of the first terms of an arithmetic series to find the sum. [reveal-answer q=”fs-id1165135471078″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135471078″] [/hidden-answer] [reveal-answer q=”656521″]Show Solution[/reveal-answer] [hidden-answer a=”656521″][/hidden-answer] For the following exercises, use the formula for the sum of the first terms of a geometric series to find the partial sum. for the series for the series [reveal-answer q=”fs-id1165137530296″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137530296″] [/hidden-answer] [reveal-answer q=”fs-id1165135496397″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135496397″] [/hidden-answer] For the following exercises, find the sum of the infinite geometric series. [reveal-answer q=”fs-id1165137535603″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137535603″] [/hidden-answer] [reveal-answer q=”fs-id1165135333829″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165135333829″] [/hidden-answer] For the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of deposits, and the interest rate. Deposit amount: total deposits: interest rate: compounded monthly Deposit amount: total deposits: interest rate: compounded monthly [reveal-answer q=”fs-id1165137705498″]Show Solution[/reveal-answer] [hidden-answer a=”fs-id1165137705498″]$3,705.42

Deposit amount: total deposits: interest rate: compounded quarterly

Deposit amount: total deposits: interest rate: compounded semi-annually

#### Real-World Applications

Keisha devised a week-long study plan to prepare for finals. On the first day, she plans to study forhour, and each successive day she will increase her study time byminutes. How many hours will Keisha have studied after one week?

A boulder rolled down a mountain, traveling 6 feet in the first second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds?

420 feet

A scientist places 50 cells in a petri dish. Every hour, the population increases by 1.5%. What will the cell count be after 1 day?

A pendulum travels a distance of 3 feet on its first swing. On each successive swing, it travels the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?

12 feet

Rachael deposits \$1,500 into a retirement fund each year. The fund earns 8.2% annual interest, compounded monthly. If she opened her account when she was 19 years old, how much will she have by the time she is 55? How much of that amount will be interest earned?

### Glossary

annuity
an investment in which the purchaser makes a sequence of periodic, equal payments
arithmetic series
the sum of the terms in an arithmetic sequence
diverge
a series is said to diverge if the sum is not a real number
geometric series
the sum of the terms in a geometric sequence
index of summation
in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation
infinite series
the sum of the terms in an infinite sequence
lower limit of summation
the number used in the explicit formula to find the first term in a series
nth partial sum
the sum of the firstterms of a sequence
series
the sum of the terms in a sequence
summation notation
a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series
upper limit of summation
the number used in the explicit formula to find the last term in a series