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Mathematical Expression Editor
We find the sum of a geometric series.
1 Definition of Infinite 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.
2 Geometric Series
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.
(problem 2a) Write the first four terms of the infinite series and answer the questions
below:
True or false: this is a geometric series TrueFalse
(problem 2b) Write the first four terms of the infinite series and answer the questions
below:
True or false: this is a geometric series TrueFalse
3 Convergence
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.
(problem 3a) Does the geometric series converge or diverge? The common ratio is The series
convergesdiverges
(problem 3b) Does the geometric series converge or diverge? The common ratio is The series
convergesdiverges
(problem 3c) Does the geometric series converge or diverge? The common ratio is The series
convergesdiverges
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.
(problem 4a) Does the geometric series converge or diverge? The common ratio is The series
convergesdiverges
(problem 4b) Does the geometric series converge or diverge? The common ratio is The series
convergesdiverges
example 5 Does the geometric series converge or diverge? 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
convergesdiverges
4 Sum of a Geometric Series
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 : \begin{align*} S_N - rS_N &= \left (a + ar + \cdots + ar^{N-1} + ar^N\right ) - \left (ar + ar^2 + \cdots + ar^N + ar^{N+1} \right )\\ &= a - ar^{N+1}. \end{align*}
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.
Sum of a Geometric Series If , then the sum of the geometric series is given by That
is, if , then
example 6 Find the sum of the geometric series The first term is , and the common
ratio is . The sum of the series is
(problem 6a) Find the sum of the geometric series The first term is The common ratio is The sum is
(problem 6b) Find the sum of the geometric series The first term is The common ratio is The sum is
(problem 6c) Find the sum of the geometric series The first term is The common ratio is The sum 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
(problem 7a) Find the sum of the geometric series The first term is The common ratio is The sum is
(problem 7b) Find the sum of the geometric series The first term is The common ratio is The sum is
(problem 7c) Find the sum of the geometric series The first term is The common ratio is The sum 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
(problem 8) Write the repeating decimal as a ratio of integers.
5 Video Lesson
Here is a detailed, lecture style video on Geometric Series: