1Use E=1to answer the parts below.(2n+1)2A. How many terms are needed to make sure that the sum is accurate to within 0.00005?

Answered: How many terms are needed to make sure…

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
Use E=1
to answer the parts below.
(2n+1)2
A. How many terms are needed to make sure that the sum is accurate to within 0.00005?

Expert Solution

Step 1

Given:

$\sum_{n = 1}^{\infty} \frac{1}{{(2 n + 1)}^{2}}$

To find:

How many terms are needed to make sure that sum is accurate to within 0.00005

Step 2

To find smallest n such that,

$\frac{1}{{(2 n + 1)}^{2}} < 0.00005$

Take reciprocal of both sides,

Since, both the side are positive, the inequality reverses.

${(2 n + 1)}^{2} > \frac{1}{0.00005}$

${(2 n + 1)}^{2} > 20000$

Check by error and trial:

$n = 1,$

$a_{1} = {(2 (1) + 1)}^{2} = 9$

$n = 2,$

$a_{2} = {(2 (2) + 1)}^{2} = 25$

$n = 3,$

$a_{3} = {(2 (3) + 1)}^{2} = 49$

$n = 6,$

$a_{6} = {(2 (6) + 1)}^{2} = 169$

$n = 7,$

$a_{7} = {(2 (7) + 1)}^{2} = 225$

$n = 9,$

$a_{9} = {(2 (9) + 1)}^{2} = 361$

and so...

$n = 65,$

$a_{65} = {(2 (65) + 1)}^{2} = 17161$

$n = 70,$

$a_{70} = {(2 (70) + 1)}^{2} = 19881$

$n = 71,$

$a_{71} = {(2 (71) + 1)}^{2} = 20449 > 20000$

Thus, $n = 71$ is the smallest number for which inequality is satisfied.

Hence, $n = 71$ is the smallest number for which the error is less than 0.00005.

The first 71 terms are needed to make sure that sum is accurate to within 0.00005

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