Chapter 7: Q. 44 (page 653)

Use any convergence tests to determine whether the series converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.
$\sum_{k = 1}^{\infty} {(- 1)}^{k} e^{- k}$

Short Answer

Expert verified

The series is absolutely convergent.

Step by step solution

Step 1. Given information.

Consider the given question,

$\sum_{k = 1}^{\infty} {(- 1)}^{k} e^{- k}$

Step 2. Consider the ratio.

The general term of the series $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} {(- 1)}^{k} e^{- k}$ is $a_{k} = {(- 1)}^{k} e^{- k}$ .

The ratio $|\frac{a_{k + 1}}{a_{k}}|$ gives,

$|\frac{a_{k + 1}}{a_{k}}| = |\frac{{(- 1)}^{k + 1} e^{- (k + 1)}}{{(- 1)}^{k} e^{- k}}| |\frac{a_{k + 1}}{a_{k}}| = |\frac{(- 1) e^{- (k + 1)}}{e^{- k}}| |\frac{a_{k + 1}}{a_{k}}| = |\frac{1}{k}|$

Step 3. Find the value of the limit.

The value of $\lim_{k \to \infty} |\frac{a_{k + 1}}{a_{k}}|$ is given below,

$\lim_{k \to \infty} |\frac{a_{k + 1}}{a_{k}}| = \lim_{k \to \infty} |\frac{1}{e}| = \frac{1}{e}$

The value of the above limit is less than $1$ .

Thus, by ratio test for absolute convergence, the series $\sum_{k = 1}^{\infty} {(- 1)}^{k} e^{- k}$ is absolutely convergent.

Hence, the series is absolutely convergent.

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Use any convergence tests to determine whether the series converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.
$\sum_{k = 1}^{\infty} {(- 1)}^{k} e^{- k}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the ratio.

Step 3. Find the value of the limit.

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Use any convergence tests to determine whether the series converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.∑k=1∞-1ke-k

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider the ratio.

Step 3. Find the value of the limit.

One App. One Place for Learning.

Most popular questions from this chapter

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Use any convergence tests to determine whether the series converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.
$\sum_{k = 1}^{\infty} {(- 1)}^{k} e^{- k}$