   Chapter 11, Problem 37RE

Chapter
Section
Textbook Problem

# Use the sum of the first eight terms to approximate the sum of the series ∑ n = 1 ∞ ( 2 + 5 n ) − 1 . Estimate the error involved in this approximation.

To determine

To approximate: The sum of the series n=1(2+5n)1 by using the first 8 terms and estimate the error.

Explanation

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=112+5n .

Since 12+5n<15n .

Compute the sum of the first 8 terms.

s8=n=1812+5n=12+51+12+52+12+53+12+54+12+55+12+56+12+57+12+580.18976224

Here, it is observed that the sum of the first 8 terms of the given series is approximately 0.18976224, 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=112+5nn=1812+5n

Therefore, the required sum of the series is approximately 0

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