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Chapter 7: Q 55. (page 615)

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{2^{k + 2}}{5^{k - 1}}$ converges or diverges. Give the sum of the convergent series.

Short Answer

Expert verified

The series ${鈭�}_{k = 0}^{鈭�} \frac{2^{k + 2}}{5^{k - 1}}$ converges to $\frac{100}{3}$ .

Step by step solution

01

Step 1. Given information.

Given a series ${鈭�}_{k = 0}^{鈭�} \frac{2^{k + 2}}{5^{k - 1}}$ .

02

Step 2. Find if the series converges or not.

The series ${鈭�}_{k = 0}^{鈭�} \frac{2^{k + 2}}{5^{k - 1}}$ can be expressed as ${鈭�}_{k = 0}^{鈭�} 20 {(\frac{2}{5})}^{k}$ .

The series ${鈭�}_{k = 0}^{鈭�} 20 {(\frac{2}{5})}^{k}$ is in the standard form ${鈭�}_{k = 0}^{鈭�} c r^{k}$ for a geometric series with $c = 20$ and $r = \frac{2}{5}$ .

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

Since $r = \frac{2}{5}$ , it follows that the series role="math" localid="1648973658696" ${鈭�}_{k = 0}^{鈭�} \frac{2^{k + 2}}{5^{k - 1}}$ 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 ${鈭�}_{k = 0}^{鈭�} \frac{2^{k + 2}}{5^{k - 1}}$ converges to $\frac{20}{1 - \frac{2}{5}}$ , that is $\frac{100}{3}$ .

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Most popular questions from this chapter

Improper Integrals: Determine whether the following improper integrals converge or diverge.

${鈭�}_{1}^{鈭�} \frac{1}{x} d x .$

Find an example of a continuous function f : $[1, 鈭�) 鈫� R$ such that ${鈭�}_{1}^{鈭�} f (x) d x$ diverges and localid="1649077247585" ${鈭�}_{k = 1}^{鈭�} f (k)$ converges.

For each series in Exercises 44鈥�47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder, $R_{10}$ .

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that $R_{n} 鈮� 10^{- 6}$

${鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}}$

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.

$0.6345345 . . .$

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.

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

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