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Use the alternating series test to determine whether the series in Exercises 24–29 converge or diverge. If a series converges, determine whether it converges absolutely or conditionally.

k=0(1)k+11+k2

Short Answer

Expert verified

The seriesk=0(1)k+11+k2converges

Step by step solution

01

Step 1. Given information

k=0(1)k+11+k2

02

Step 2. Solving the series

k=0(1)k+11+k2=k=1(1)k+1akwhereak=11+k2

Substitute k=k+1inak=11+k2,

ak+1=(k+1)k+1(k+1)!

03

Step 3. Calculate the limit

limkak=limk11+k2=0

Therefore,k=0(1)k+11+k2converges absolutely.

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