(a) Approximate the sum of the series E1/n by15 terms. Estimate the error involved in this approximation. (bterms are required to ensure that the sum is accurate to withinEXAMPLE 5In both parts (a) and (b) we need to know Sf(x)dx1/x, which satisfies the conditions of the Integral Test, we havSOLUTION1dr = lim4x1= lim4t44n414n4%D11 1 11E -* S15 =-++ -+ ... +* 1.03692n=1 n515 25 35155According to the remainder estimate for the Integral Test, we h1R15 < .-dx =so the size of the error is at most 0.000004938.(b) Accuracy to within 0.0005 means that we have to find a valthat Rn < 0.0005. Since1R, <1dr%3D4n4we want14n< 0.0005. Solving this inequality, we get1n4 >50040.0005n > 4.7287We needterms to ensure accuracy to within 0.0005.

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Answered: EXAMPLE 5 (a) Approximate the sum of…

EXAMPLE 5 (a) Approximate the sum of the series 21/n by 15 terms. Estimate the error involved in this approximation. (b terms are required to ensure that the sum is accurate to within

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

(a) Approximate the sum of the series E1/n by
15 terms. Estimate the error involved in this approximation. (b
terms are required to ensure that the sum is accurate to within
EXAMPLE 5
In both parts (a) and (b) we need to know Sf(x)dx
1/x, which satisfies the conditions of the Integral Test, we hav
SOLUTION
1
dr = lim
4x
1
= lim
4t4
4n4
1
4n
4
%D
1
1 1 1
1
E -* S15 =-+
+ -+ ... +
* 1.03692
n=1 n5
15 25 35
155
According to the remainder estimate for the Integral Test, we h
1
R15 < .
-dx =
so the size of the error is at most 0.000004938.
(b) Accuracy to within 0.0005 means that we have to find a val
that Rn < 0.0005. Since
1
R, <
1
dr%3D
4n
4
we want
1
4n
< 0.0005. Solving this inequality, we get
1
n4 >
500
40.0005
n > 4.7287
We need
terms to ensure accuracy to within 0.0005.

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