   Chapter 11.7, Problem 38E

Chapter
Section
Textbook Problem

Test the series for convergence or divergence. ∑ n = 1 ∞ ( 2 n − 1 )

To determine

To test:

The series for convergence or divergence.

Explanation

1) Concept:

i) If the series has a form that is similar to a p-series or a geometric series, then one of the comparison tests should be considered. In particular, if an is a rational function or an algebraic function of n(involving roots of polynomials), then the series should be compared with a p-series.

ii) A p-series [1/np converges if p>1 and diverges if p1]

Limit Comparison Test:

an and bn are the series with positive terms

If limnanbn=c, where c is a finite number and c>0, then either both series converge or both diverge

2) Given:

n=12n-1

3) Calculation:

Here,

an=2n-1 is algebraic function of n.

By concept i) (use the limit comparison test).

Choose.

bn=1n

Then,

limnanbn=limn2n-11n

=limx21x

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