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Use the ratio test for absolute convergence to determine whether the series in Exercises 30–35 converge absolutely or diverge.
k=0(-3)k+1(2k)!

Short Answer

Expert verified

The series k=0(-3)k+1(2k)!converges absolutely.

Step by step solution

01

Step 1. Given Information.

The series:

k=0(-3)k+1(2k)!

02

Step 2. Rewrite the series.

ak=(-3)k+1(2k)!

03

Step 3. Find ak+1.

ak+1=(-3)k+1+1(2(k+1))!=(-3)k+2(2k+2)!

04

Step 4. Calculate ak+1ak.

ak+1ak=(-3)k+2(2k+2)!(-3)k+1(2k)!=(-3)k+2(2k)!(-3)k+1(2k+2)!=-3(-3)k+1(2k)!(-3)k+1(2k+2)(2k+1)(2k)!=-3(2k+2)(2k+1)=32(k+1)(2k+1)

05

Step 5. Take limits.

limkak+1ak=limk(32(k+1)(2k+1))=0

So by the ratio test, the series converges absolutely.

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