1n=2 n(In n)5is convergent.(A). According to the Remainder Estimate for the Integral Test, the error in the approximation s 2 Sn (where s is the value of the infinite sum and s, is the n-thpartial sum) is|s – Sn| <(B). Find the smallest value of n such that this upper bound is less than 0.006.п —

Given series ∑n=2∞1nlnn5 (A) S−Sn≤∫n∞1xlnx5dx Substitute lnx=u ⇒1xdx=du And limit of u is lnn to ∞…

Answered: 1 n(ln n)5 n=2 is convergent. (A).…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

1
n=2 n(In n)5
is convergent.
(A). According to the Remainder Estimate for the Integral Test, the error in the approximation s 2 Sn (where s is the value of the infinite sum and s, is the n-th
partial sum) is
|s – Sn| <
(B). Find the smallest value of n such that this upper bound is less than 0.006.
п —

Expert Solution

Step 1

Given series $\sum_{n = 2}^{\infty} \frac{1}{n {(\ln n)}^{5}}$

(A) $|S - S_{n}| \leq \int_{n}^{\infty} \frac{1}{x {(\ln x)}^{5}} d x$

Substitute $\ln x = u$

$\Rightarrow \frac{1}{x} d x = d u$

And limit of $u$ is $\ln n$ to $\infty$

Hence,

$\begin{matrix} |S - S_{n}| \leq \int_{n}^{\infty} \frac{1}{x {(\ln x)}^{5}} d x \\ \leq \int_{\ln n}^{\infty} \frac{1}{u^{5}} d u \\ \leq {[\frac{1}{- 4 u^{4}}]}_{\ln n}^{\infty} \\ \leq [0 + \frac{1}{4 {(\ln n)}^{4}}] \\ \leq \frac{1}{4 {(\ln n)}^{4}} \end{matrix}$

Therefore, $|S - S_{n}| \leq \frac{1}{4 {(\ln n)}^{4}}$

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