We find the sum of a geometric series.

1 Definition of Infinite Series

2 Geometric Series

(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

(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

(problem 3a) Does the geometric series converge or diverge?
The common ratio is
The series
converges diverges
(problem 3b) Does the geometric series converge or diverge?
The common ratio is
The series
converges diverges
(problem 3c) Does the geometric series converge or diverge?
The common ratio is
The series
converges diverges
(problem 4a) Does the geometric series converge or diverge?
The common ratio is
The series
converges diverges
(problem 4b) Does the geometric series converge or diverge?
The common ratio is
The series
converges diverges
(problem 5) Does the geometric series converge or diverge?
The common ratio is
The series
converges diverges

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.

(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
(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
(problem 8) Write the repeating decimal as a ratio of integers.

5 Video Lesson

Here is a detailed, lecture style video on Geometric Series:
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2024-09-27 13:58:02