   Chapter 11.3, Problem 12E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent.12. 1 5 + 1 7 + 1 9 + 1 11 + 1 13 + ⋯

To determine

Whether the series is convergent or divergent.

Explanation

Result used:

If the function f(x) is continuous, positive and decreasing on [1,) and let an=f(n) , then the series n=1an is divergent if and only if the improper integral 1f(x)dx is divergent.

Definition used:

The improper integral abf(x)dx is divergent if the limit does not exist.

Chain rule: d[f(x)]ndx=n[f(x)]n1f(x)

Given:

The series is 15+17+19+111+113+... .

Calculation:

The given series can be written as follows,

15+17+19+111+113+...=1(2(1)+3)+1(2(2)+3)+1(2(3)+3)+1(2(4)+3)+1(2(5)+3)+...=n=11(2n+3)=n=1(2n+3)1

Consider the function from the above series f(x)=(2x+3)1

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