Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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.
$\sum_{k = 1}^{\infty} \frac{k}{e^{k}}$

Ans: The series $\sum_{k = 1}^{\infty} \frac{k}{e^{k}}$ is convergent.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

given,

$\sum_{k = 1}^{\infty} \frac{k}{e^{k}}$

Step 2. The objective is to explain why the integral test is used to determine the convergence or divergence of the series and use the test to determine the convergence or divergence of the series.

Consider function $f (x) = x e^{- x}$ .

The first derivative of the function $f (x) = x e^{- x}$ is:

$\begin{aligned} f^{'} (x) = e^{- x} - x e^{- x} \\ = (1 - x) e^{- x} \end{aligned}$

The derivative is negative for all $x > 1$ . Therefore the function is decreasing.

The function $f (x) = x e^{- x}$ is continuous, decreasing, with positive terms. Therefore, all the conditions of the integral test are fulfilled. So, the integral test is applicable.

Step 3. To evaluate the integral, integrate by parts.

Therefore,

$\begin{aligned} \int_{x = 1}^{\infty} x e^{- x} d x = lim_{k \to \infty} \int_{x = 1}^{k} x e^{- x} d x \\ = lim_{k \to \infty} {[- x e^{- x} - (e^{- x})]}_{x = 1}^{k} (Integrate) \\ = lim_{k \to \infty} {[- x e^{- x} - e^{- x}]}_{x = 1}^{k} \\ = lim_{k \to \infty} [- k e^{- k} - e^{- k} + 2 e^{- 1}] (Substitution) \\ = 2 e^{- 1} (Take limit) \end{aligned}$

Step 4. Thus, the value of the integral is ∫x=1∞ xe−xdx=2e.

The integral converges. Therefore, the series $\sum_{k = 1}^{\infty} \frac{k}{e^{k}}$ is convergent.

Hence, by integral test, the series $\sum_{k = 1}^{\infty} \frac{k}{e^{k}}$ is convergent.

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Short Answer

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.
$\sum_{k = 1}^{\infty} \frac{k}{e^{k}}$

Step by Step Solution

Step 1. Given information.

Step 2. The objective is to explain why the integral test is used to determine the convergence or divergence of the series and use the test to determine the convergence or divergence of the series.

Step 3. To evaluate the integral, integrate by parts.

Step 4. Thus, the value of the integral is ∫x=1∞ xe−xdx=2e.

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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. ∑k=1∞ kek

Step by Step Solution

Step 1. Given information.

Step 2. The objective is to explain why the integral test is used to determine the convergence or divergence of the series and use the test to determine the convergence or divergence of the series.

Step 3. To evaluate the integral, integrate by parts.

Step 4. Thus, the value of the integral is ∫x=1∞ xe−xdx=2e.

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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.
$\sum_{k = 1}^{\infty} \frac{k}{e^{k}}$