   Chapter 11.6, Problem 17E

Chapter
Section
Textbook Problem

# Use the Ratio Test to determine whether the series is convergent or divergent.17. ∑ n = 1 ∞ cos ( n π / 3 ) n !

To determine

Whether the series is convergent or divergent.

Explanation

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.”

Theorem:

If a series an is absolutely convergent, it is convergent.

Calculation:

The given series n=1an=n=1cos(nπ3)n!.

Here, the nth term is, an=cos(nπ3)n!.

Thus, the (n+1) th term is, an+1=cos((n+1)π3)(n+1)!.

Obtain the limit of |an+1an|

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