Prove the absolute convergence of the series?

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1 Answer
Dec 30, 2017

The Ratio Test is the best thing to use here.

Explanation:

To prove that sum_{n=1}^{infty}((-100)^{n})/(n!) converges absolutely, we must show that sum_{n=1}^{infty}|((-100)^{n})/(n!)|=sum_{n=1}^{infty}(100^{n})/(n!) converges.

This is best done by the Ratio Test. Let a_{n}=100^{n}/(n!). Then a_{n+1}=(100*100^{n})/((n+1)*n!) and a_{n+1}/a_{n}=100/(n+1)->0=L as n->infty.

Since L<1, it follows that sum_{n=1}^{infty}a_{n}=sum_{n=1}^{infty}(100^{n})/(n!) converges. Hence, sum_{n=1}^{infty}((-100)^{n})/(n!) converges absolutely.