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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)k+1k1002k

Short Answer

Expert verified

The series k=1(-1)k+1k1002k converges absolutely.

Step by step solution

01

Step 1. Given Information.

The series:

k=1(-1)k+1k1002k

02

Step 2. By Alternating Series Test.

According to the Alternating Series Test, the sequence ak+1<akfor every . Then the alternating series ak+1,akboth converges.

03

Step 3. Find ak+1.

ak=k1002kak+1=(k+1)1002k+1ak+1<ak

So the sequence is monotonic decreasing sequence.

04

Step 4. Find limk→∞ak.

limkak=limkk1002k=0

So the series converges.

05

Step 5. Use Ratio test.

limkak+1ak=limk(-1)k+1(k+1)1002k+1(-1)kk1002k=limk2k(k+1)100k1002k+1=limk(k+1)1002k100=12<1

By Ratio test, the series converges absolutely.

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