Convergence of the function series $\sum _{n=1}^{\infty}\dfrac{(nx)^n}{n!}$ [duplicate]
Solution 1:
There is:
\begin{align} \frac{(n + 1)^{n + 1}/(n + 1)!}{n^n/n!} &= \frac{(n + 1)^{n}(n+1)/(n + 1)!}{n^n/n!} \\ &= \frac{(n + 1)^{n}/n!}{n^n/n!} \\ &= \frac{(n + 1)^{n}}{n^n} \\ &= \left( 1 + \frac1n \right)^n \\ &\to e. \end{align}