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Chapter 7: Q 56. (page 615)

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

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given information.

Given a series ${鈭�}_{k = 0}^{鈭�} \frac{4^{k + 1}}{3^{2 k}}$ .

02

Step 2. Find if the series converges or not.

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

The series ${鈭�}_{k = 0}^{鈭�} 4 {(\frac{4}{9})}^{k}$ is in the standard form role="math" localid="1648976707514" ${鈭�}_{k = 0}^{鈭�} c r^{k}$ 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" ${鈭�}_{k = 0}^{鈭�} \frac{4^{k + 1}}{3^{2 k}}$ converges.

03

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

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

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

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Most popular questions from this chapter

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

${鈭�}_{k = 2}^{鈭�} 鈥� \frac{1}{k (\ln 鈦� k)^{2}}$

Prove Theorem 7.24 (a). That is, show that if c is a real number and ${鈭�}_{k = 1}^{鈭�} a_{k}$ is a convergent series, then ${鈭�}_{k = 1}^{鈭�} c a_{k} = c {鈭�}_{k = 1}^{鈭�} a_{k}$ .

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{3}{2^{k}}$ converges or diverges. Give the sum of the convergent series.

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{1 + k}{k^{2}}$

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.305130513051 . . .$

See all solutions

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