This is question.I need to sovle exacly the same problem like #5 but teacher told us that in the proof we should use the fact that (1+(1/n))^n goes to e.
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$\displaystyle \begin{aligned}\lim_{n\to\infty}\left|\frac{(n+1)^ {n+1}x^{n+1}}{(n+1)!}\cdot\frac{n!}{n^nx^n}\right| &=|x|\lim_{n\to\infty}\left|\frac{(n+1)^{n}}{n^n}\ right|\\
&=|x|\lim_{n\to\infty}\left|\left(1+\frac{1}{n}\ri ght)^n\right|\\
&=|x|e<1\end{aligned}$