Please give detailed answer. Thanks.1dx dy.x(1+ y?)5. Compute the value of the seriesn=1n+1

Answered: re 1 dx dy. x(1+ y²) Compute the value…

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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Please give detailed answer. Thanks.
1
dx dy.
x(1+ y?)
5. Compute the value of the series
n=1
n+1

Expert Solution

Step 1

To find the value of given series first we solve the integration.

$\sum_{n = 1}^{\infty} \int_{\frac{1}{n + 1}}^{\frac{1}{n}} \int_{1}^{e} \frac{1}{x (1 + y^{2})} d x d y = \sum_{n = 1}^{\infty} \int_{\frac{1}{n + 1}}^{\frac{1}{n}} (\int_{1}^{e} \frac{1}{x} d x) \frac{1}{(1 + y^{2})} d y S i n c e w e k n o w t h a t \int_{1}^{e} \frac{1}{x} d x = \ln x, t h e n = \sum_{n = 1}^{\infty} \int_{\frac{1}{n + 1}}^{\frac{1}{n}} ({[\ln x]}_{1}^{e}) \frac{1}{(1 + y^{2})} d y$

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