   Chapter 11.3, Problem 39E

Chapter
Section
Textbook Problem

Estimate ∑ n = 1 ∞ ( 2 n + 1 ) − 6 correct to five decimal places.

To determine

To estimate:

The sum of the series n=12n+1-6 correct to four decimal places.

Explanation

1) Concept:

Use integral test and remainder estimate for integral test

2) Formula:

a) Hypothesis for integral test:

Suppose f is a continuous, positive, decreasing function on [1, ] and let an=f(n). Then the series n=1an is convergent if and only if the improper integral 1f(x)dx is convergent.

b) Remainder Estimate for integral test:

If fk=ak, where f is continuous, positive decreasing function for xn and an is convergent. If Rn=s-sn, then

n+1f(x)dxRn nf(x)dx

3) Calculation:

The function fx=2x+1-6 is continuous, positive, decreasing function on [1, ]

Therefore, hypothesis of Integral Test satisfied so Integral test applies

Applying Remainder Estimate for integral test,

Rn n2x+1-6dx

Simplify,

=limt2x+1-5(-5)(2)nt

=limt-1102x+15nt

=limt-1102t+15+1102n+15

Apply limit separately,

=limt-1102t+15+limtȡ

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