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Explain why the ratio test does not work in determining the convergence or divergence of the series k=1k!(k+2)!. What test would be more effective to analyze this series?

Short Answer

Expert verified

Hence proved that the ratio test will be inconclusive becauseL=1and Convergence and Divergence test can be used.

Step by step solution

01

Step 1. Given information.

We are givenk=1k!(k+2)!.

02

Step 2. Ratio Test.

On using Ratio Test,

ak+1=(k+1)!(k+3)!ak+1ak=(k+1)!(k+3)!k!(k+2)!ak+1ak=(k+1)k!(k+2)!(k+3)(k+2)!k!=k+1k+3limkak+1ak=limkk+1k+3=limkk1+1kk1+3kL=1Therefore,thetestisinconclusive.

03

Step 3. Convergence and Divergence test.

Now,

k=1ak=k=1k!(k+2)!andk=1bk=k=11k2Clearly,0k!(k+2)!1k2.Now,k=1bk-k=11k2is of the formk=1bk-k=11kpwhich ie apeeries.Here,p=2>1.Hence, the seriesk=1bk=k=11k2converges.

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