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

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

Advanced Engineering Mathematics
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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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?
Transcribed Image Text: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:

∑n=1∞ 1(2n+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,

 1(2n+1)2<0.00005

Take reciprocal of both sides,

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

 (2n+1)2>10.00005

 (2n+1)2>20000

Check by error and trial:

n=1,

a1= (2(1)+1)2=9

n=2,

a2= (2(2)+1)2=25

n=3,

a3= (2(3)+1)2=49

n=6,

a6= (2(6)+1)2=169

n=7,

a7= (2(7)+1)2=225

n=9,

a9= (2(9)+1)2=361

 and so...

n=65,

a65= (2(65)+1)2=17161

n=70,

a70= (2(70)+1)2=19881

n=71,

a71= (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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