Chapter 12: Q. 23 (page 838)

In Problems 22–25, determine whether each infinite geometric series converges or diverges. If it converges, find its sum
$2 - 1 + \frac{1}{2} - \frac{1}{4} + . . . . .$

Short Answer

Expert verified

The geometric series converges.

Its sum is: $\frac{4}{3}$

Step by step solution

Step 1. Given information.

The series is :

2 - 1 + \frac{1}{2} - \frac{1}{4} + . . . . .

Step 2. To find common ratio and first term.

First term of the series is: $a = 2$

Common ratio is:

$r = \frac{- 1}{2}$

$|r| = \frac{1}{2}$

Since $|r| < 1$

Hence given series converges.

Step 3. To find sum of infinite terms.

$\begin{array}{rcl} S_{\infty} & = & \frac{2}{1 - (- \frac{1}{2})} \\ = & \frac{2}{1 + \frac{1}{2}} \\ = & \frac{2}{\frac{3}{2}} \\ = & \frac{4}{3} \end{array}$

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In Problems 22–25, determine whether each infinite geometric series converges or diverges. If it converges, find its sum
$2 - 1 + \frac{1}{2} - \frac{1}{4} + . . . . .$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. To find common ratio and first term.

Step 3. To find sum of infinite terms.

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In Problems 22–25, determine whether each infinite geometric series converges or diverges. If it converges, find its sum2-1+12-14+.....

Short Answer

Step by step solution

Step 1. Given information.

Step 2. To find common ratio and first term.

Step 3. To find sum of infinite terms.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Mechanics Maths

Applied Mathematics

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

In Problems 22–25, determine whether each infinite geometric series converges or diverges. If it converges, find its sum
$2 - 1 + \frac{1}{2} - \frac{1}{4} + . . . . .$