Chapter 12: Q. 25 (page 838)

In Problems 22–25, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k = 1}^{\infty} 4 {(\frac{1}{2})}^{k - 1}$

Short Answer

Expert verified

The geometric series converges.

Its sum is: $8$

Step by step solution

Step 1. Given information.

Geometric series:

$\sum_{k = 1}^{\infty} 4 {(\frac{1}{2})}^{k - 1}$

Step 2. To find common ratio and first term.

By comparing $\sum_{k = 1}^{\infty} 4 {(\frac{1}{2})}^{k - 1}$ to $\sum_{k = 1}^{\infty} a . r^{k - 1}$

We get first term: $a = 4$

Common ratio: $r = \frac{1}{2}$

Since common ratio is less than 1.

Hence the series converges.

Step 3. To find sum of infinite terms.

$\begin{array}{rcl} S_{\infty} & = & \frac{4}{1 - \frac{1}{2}} \\ = & \frac{4}{\frac{1}{2}} \\ = & 4 \times 2 \\ = & 8 \end{array}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problems 22–25, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k = 1}^{\infty} 4 {(\frac{1}{2})}^{k - 1}$

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

Mechanics Maths

Applied Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Decision Maths

Pure Maths

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.∑k=1∞412k-1

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

Mechanics Maths

Applied Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Decision Maths

Pure Maths

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.
$\sum_{k = 1}^{\infty} 4 {(\frac{1}{2})}^{k - 1}$