# Chapter 11 - Infinite Series - Chapter Review Exercises - Page 592: 90

Converges

#### Work Step by Step

Given $$\sum_{n=1}^{\infty} \frac{e^{n}}{n !}$$ By using the Ratio test \begin{align*} \rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\ &=\lim _{n \to \infty}\frac{e^{n+1}}{(n+1) !} \frac{n !} {e^{n}}\\ &=\lim _{n \to \infty}\frac{ee^{n}}{(n+1)n !} \frac{n !} {e^{n}}\\ &=\lim _{n \to \infty}\frac{e}{(n+1)}\\ &=0\lt 1 \end{align*} Thus the given series converges.

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