Question

Prove the absolute convergence of the series?

Answer 1

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.