   Chapter 11.4, Problem 35E

Chapter
Section
Textbook Problem

# Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.35. ∑ n = 1 ∞ 5 − n cos 2 n

To determine

To approximate: The sum of the series by using the first 10 terms and estimate the error.

Explanation

Given:

The series is n=15ncos2n.

Result used: Remainder Estimate for the Integral Test

If the function f(k)=ak, where f is a continuous, positive and decreasing function for xn and an is convergent and Rn=ssn, then n+1f(x)dxRnnf(x)dx.

Calculation:

The given series is n=15ncos2n.

cos2n1  cos2n5n15n

Expand the first ten terms, s10=n=1105ncos2n of the given series.

s10=cos2(1)51+cos2(2)52+cos2(3)53+...+cos2(10)510=(0.058385+0.006927+0.007840+0.000683+0.000025+0.000059+0.000007+0.00000+0.000000+0.000000)=0.0739260.07393

Here, it is observed that the sum of the first 10 terms of the given series is approximately 0.07393, which is added to the next proceeding terms up to large value of n then also there is no change in the sum.

That is,

n=15ncos2nn=1105ncos2n

Therefore, the required sum of the series is approximately 0.07393.

Thus, by the Result stated above, the error R10=ss10105xcos2xdx

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