True or false: this is a geometric series TrueFalse

The first term is

The common ratio is

We find the sum of a geometric series.

Infinite Series We define If this limit is a finite number, then we say that the infinite
series **converges**. If this limit is not finite or if it does not exist, then we say that the
infinite series **diverges**.

The sum, , is called the partial sum of the infinite series . Thus, in words, we define
an infinite series to be the limit of its partial sums. This is similar to an improper
integral which is also defined in terms of a limit.

Geometric Series A **geometric series** is an infinite series of the form The parameter
is called the **common ratio**.

example 1 Consider the series: Write out the first four terms of the series. If the
series is geometric, find the common ratio.

The first four terms are

The series is geometric and the common ratio is .

(problem 1a) Write the first four terms of the infinite series and answer the questions
below:

True or false: this is a geometric series TrueFalse

The first term is

The common ratio is

(problem 1b) Write the first four terms of the infinite series and answer the questions
below:

True or false: this is a geometric series TrueFalse

The first term is

The common ratio is

example 2 Consider the series: Write out the first four terms of the series. If the
series is geometric, find the common ratio.

The first four terms are The ratio of the second term to the first is but the ratio of the third term to the second is . The series is not geometric because it fails to have a common ratio.

Convergence of Geometric Series The geometric series converges if and only if the
common ratio satisfies the condition . This condition can also be written as
.

example 3 Does the geometric series converge or diverge?

The common ratio of this geometric series is . Since , the series converges.

The common ratio of this geometric series is . Since , the series converges.

(problem 3a) Does the geometric series converge or diverge?

The common ratio is

The series

The common ratio is

The series

converges diverges

(problem 3b) Does the geometric series converge or diverge?

The common ratio is

The series

The common ratio is

The series

converges diverges

(problem 3c) Does the geometric series converge or diverge?

The common ratio is

The series

The common ratio is

The series

converges diverges

example 4 Does the geometric series converge or diverge?

The common ratio of this geometric series is . Since is not between and , the series does not converge, i.e., it diverges.

The common ratio of this geometric series is . Since is not between and , the series does not converge, i.e., it diverges.

(problem 4a) Does the geometric series converge or diverge?

The common ratio is

The series

The common ratio is

The series

converges diverges

(problem 4b) Does the geometric series converge or diverge?

The common ratio is

The series

The common ratio is

The series

converges diverges

example 5 Does the geometric series converge or diverge?

The common ratio of this geometric series is . Since , the series converges.

The common ratio of this geometric series is . Since , the series converges.

(problem 5) Does the geometric series converge or diverge?

The common ratio is

The series

The common ratio is

The series

converges diverges

Consider the geometric series where (so that the series converges). The partial sum of this series is given by Multiply both sides by : Now subtract from :

We can factor out on the left side and then divide by to obtain We can now compute the sum of the geometric series by taking the limit as : We present this formula in the theorem below.example 6 Find the sum of the geometric series The first term is , and the common
ratio is . The sum of the series is

example 7 Find the sum of the geometric series The first term is , and the common
ratio is . The sum of the series is

example 8 Write the repeating decimal as a ratio of integers.

First, note that we can write this repeating decimal as an infinite series: Wring these decimals as fractions, we have This is a convergent geometric series with first term, and common ratio . The sum is

First, note that we can write this repeating decimal as an infinite series: Wring these decimals as fractions, we have This is a convergent geometric series with first term, and common ratio . The sum is

Here is a detailed, lecture style video on Geometric Series:

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