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Chapter 7: Q. 24 (page 625)

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

Short Answer

Expert verified

Ans: The series ${鈭�}_{k = 1}^{鈭�} 鈥� \frac{k}{e^{k}}$ is convergent.

Step by step solution

01

Step 1. Given information.

given,

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

02

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:

$\begin{aligned} f^{鈥�} (x) = e^{鈭� x} 鈭� x e^{鈭� x} \\ = (1 鈭� x) e^{鈭� x} \end{aligned}$

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.

03

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

Therefore,

$\begin{aligned} {鈭�}_{x = 1}^{鈭�} 鈥� x e^{鈭� x} d x = lim_{k 鈫� 鈭�} 鈥� {鈭�}_{x = 1}^{k} 鈥� x e^{鈭� x} d x \\ = lim_{k 鈫� 鈭�} 鈥� {[鈭� x e^{鈭� x} 鈭� (e^{鈭� x})]}_{x = 1}^{k} (Integrate) \\ = lim_{k 鈫� 鈭�} 鈥� {[鈭� x e^{鈭� x} 鈭� e^{鈭� x}]}_{x = 1}^{k} \\ = lim_{k 鈫� 鈭�} 鈥� [鈭� k e^{鈭� k} 鈭� e^{鈭� k} + 2 e^{鈭� 1}] (Substitution) \\ = 2 e^{鈭� 1} (Take limit) \end{aligned}$

04

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

The integral converges. Therefore, the series ${鈭�}_{k = 1}^{鈭�} 鈥� \frac{k}{e^{k}}$ is convergent.

Hence, by integral test, the series ${鈭�}_{k = 1}^{鈭�} 鈥� \frac{k}{e^{k}}$ is convergent.

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Most popular questions from this chapter

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31鈥�48 converge or diverge. Explain how the series meets the hypotheses of the test you select.

${鈭�}_{k = 1}^{鈭�} \frac{\sqrt{k}}{k^{2} + 3}$

Find the values of x for which the series ${鈭�}_{K = 0}^{鈭�} {(\sin x)}^{k}$ converges.

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 that if ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges to L and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges to M , then the series ${鈭�}_{k = 1}^{鈭�} (a_{k} + b_{k}) = L + M$ .

Given that $a_{0} = - 3, a_{1} = 5, a_{2} = - 4, a_{3} = 2$ and ${鈭� a_{k}}_{k = 2}^{鈭�} = 7$ , find the value of ${鈭� a_{k}}_{k = 4}^{鈭�}$ .

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