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Let $S = \sum_{k=1}^{\infty} \dfrac{k!}{k^k}$. Then $|\tfrac{a_{n+1}}{a_n}| = \dfrac{(k+1) k^k}{(k+1)^{k+1}} = \Big( \dfrac{k}{k+1} \Big)^k = \Big( 1 + \dfrac{(-1)}{k+1} \Big)^k \to 1/e$ as $k \to \infty$.
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Wed, 08 Nov 2017 04:36 GMT