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Conditional and absolute convergence: For each of the series that follow, determine whether the series converges absolutely, converges conditionally, or diverges. Explain the criteria you are using and why your conclusion is valid.

k=1(-1)k1k(1+k)

Short Answer

Expert verified

The series k=1(-1)k1k(1+k)converges conditionally.

Step by step solution

01

Step 1. Given Information.

The series:

k=1(-1)k1k(1+k)

02

Step 2. By Alternating Series Test.

According to the Alternating Series Test, the sequence ak+1<akfor every k1,limkak=0. Then the alternating series ak,ak+1both converges.

03

Step 3. Find ak+1.

ak+1=1(k+1)(k+1+1)=1(k+1)(k+2)ak+1<ak

So the sequence is monotonic decreasing sequence.

04

Step 4. Find limk→∞ak.

limkak=limk1k(k+1)=0

So the series converges.

05

Step 5. Find bk.

To find if the series converges absolutely or conditionally.

bk=1kwhich is a dominant term.

limkakbk=limk1k(k+1)1k=limkkk2(1+1k)=limk11+1k=0

So the series bkdiverges.

And by Limit Comparison Test, the series converges conditionally.

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