   Chapter 11.4, Problem 40E

Chapter
Section
Textbook Problem

# (a) Suppose that Σ an and Σ bn are series with positive terms and Σ bn is convergent. Prove that if lim n → ∞ a n b n = 0 then Σ an is also convergent. (b) Use part (a) to show that the series converges. (i) ∑ n = 1 ∞ ln n n 3 (ii) ∑ n = 1 ∞ ln n n   e n

(a)

To determine

To prove: The series an convergent.

Explanation

Result used:

“Suppose that an and bn are the series with positive terms,

(a) If bn is convergent and anbn for all n , then an is also convergent.

(b) If bn is divergent and anbn for all n , then an is also divergent.”

Given:

The series an and bn is convergent, limnanbn=0 . (1)

Proof:

Consider the convergent series an and bn

(b) (i)

To determine

To show: The series n=1lnnn3 converges.

(ii)

To determine

To show: The series n=1lnnnen converges.

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