   Chapter 11.6, Problem 40E

Chapter
Section
Textbook Problem

# A series Σ an is defined by the equations a 1 = 1   a n + 1 = 2 + cos n n a n Determine whether Σ an converges or diverges.

To determine

Whether the series converges or diverges.

Explanation

Given:

The terms of a an series are defined recursively by the equations a1=1 and an+1=2+cosnnan .

Theorem used: Squeeze theorem

Let f, g and h be functions such that for all x[a,b] (except possibly at the limit point c), f(x)g(x)h(x) and limxcf(x)=limxcg(x)=L then for any acb , limxch(x)=L .

Result used: The Ratio Test

(i) If limn|an+1an|=L<1 , then the series n=1an is absolutely convergent (and therefore convergent.)

(ii) If limn|an+1an|=L>1 or limn|an+1an|= , then the series n=1an is divergent.

(ii) If limn|an+1an|=1 , the Ratio Test inconclusive; that is, no conclusion can be drawn about the convergence or divergence of n=1an .

Calculation:

Consider the terms of a an series are defined recursively by the equations a1=2 and an+1=2+cosnnan .

an+1=2+cosnnan|an+1an|=|2+cosnn|

Obtain the limit of |an+1an|

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