Chapter 7: Q 56. (page 615)

Determine whether the series $\sum_{k = 0}^{\infty} \frac{4^{k + 1}}{3^{2 k}}$ converges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series $\sum_{k = 0}^{\infty} \frac{4^{k + 1}}{3^{2 k}}$ converges to $\frac{36}{5}$ .

Step by step solution

Step 1. Given information.

Given a series $\sum_{k = 0}^{\infty} \frac{4^{k + 1}}{3^{2 k}}$ .

Step 2. Find if the series converges or not.

The series $\sum_{k = 0}^{\infty} \frac{4^{k + 1}}{3^{2 k}}$ can be expressed as $\sum_{k = 0}^{\infty} 4 {(\frac{4}{9})}^{k}$ .

The series $\sum_{k = 0}^{\infty} 4 {(\frac{4}{9})}^{k}$ is in the standard form role="math" localid="1648976707514" $\sum_{k = 0}^{\infty} c r^{k}$ for a geometric series with $c = 4$ and $r = \frac{4}{9}$ .

The geometric series converges if and only if $|r| < 1$ .

Since $r = \frac{4}{9}$ , it follows that the series role="math" localid="1648976736831" $\sum_{k = 0}^{\infty} \frac{4^{k + 1}}{3^{2 k}}$ converges.

Step 3. Find the value to which the series converges.

If the geometric series $\sum_{k = 0}^{\infty} c r^{k}$ converges, it converges to $\frac{c}{1 - r}$ .

So, the series $\sum_{k = 0}^{\infty} \frac{4^{k + 1}}{3^{2 k}}$ converges to $\frac{4}{1 - \frac{4}{9}}$ , that is $\frac{36}{5}$ .

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.

Determine whether the series $\sum_{k = 0}^{\infty} \frac{4^{k + 1}}{3^{2 k}}$ converges or diverges. Give the sum of the convergent series.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find if the series converges or not.

Step 3. Find the value to which the series converges.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Pure Maths

Mechanics Maths

Discrete Mathematics

Calculus

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Determine whether the series ∑k=0∞4k+132kconverges or diverges. Give the sum of the convergent series.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find if the series converges or not.

Step 3. Find the value to which the series converges.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Pure Maths

Mechanics Maths

Discrete Mathematics

Calculus

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Determine whether the series $\sum_{k = 0}^{\infty} \frac{4^{k + 1}}{3^{2 k}}$ converges or diverges. Give the sum of the convergent series.