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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+1sin(πk)

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given Information.

The series:

k=1(-1)k+1sin(πk)

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=sin(πk)ak+1=sin(πk+1)ak+1<ak

So the series is monotonic decreasing sequence.

04

Step 4. Find limk→∞ak.

limkak=limksin(πk)=0

So the series converges.

05

Step 5. Find bk.

To find if the series converges absolutely or conditionally bk=sinπkwhich is a dominant term.

limkbkis divergent by p-series test.

So the given series is conditionally convergent.

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