   Chapter 11.4, Problem 42E

Chapter
Section
Textbook Problem

Give an example of a pair of series ∑ a n and ∑ b n with positive terms where lim n → ∞ ( a n / b n ) = 0 and ∑ b n diverges, but ∑ a n converges. (Compare with Exercise 40.)

To determine

To give:

An example of a series an and bn such that limnanbn=0 and bn diverges but an converges.

Explanation

1) Concept:

an and bn are series with positive terms then if

limnanbn=c Here c >0 is a finite number. Then both series an and bn both converge or diverge.

2) Calculation:

The p-series test states that the series 1np is convergent if p>1

Therefore, the series 1n3 is convergent, and the series 1n  is divergent So let us consider the series an=1n3  and  bn=1n

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