Answer by Diger for Convergence of...
By CS and Stirling we have $$\left|\sum_{n=1}^{\infty}\frac{(2i)^{n}\cdot n!}{n^{n}}\right| \leq \sum_{n=1}^{\infty}\frac{2^{n}\cdot n!}{n^{n}} \leq \sum_{n=1}^{\infty}\frac{2^{n}\cdot \sqrt{2\pi n} \,...
View ArticleAnswer by Bachamohamed for Convergence of...
\begin{align*}\sum_{n=1}^{\infty}\frac{(2i)^{n}{n!}}{n^n}&=\sum_{n=1}^{\infty}\frac{(2e^{i\frac{\pi}{2}})^{n}{n!}}{n^n}\\&...
View ArticleConvergence of $\displaystyle\sum_{n=1}^{\infty}\frac{(2i)^{n}\cdot n!}{n^{n}}$
I have to show the series$$\sum_{n=1}^{\infty}\frac{(2i)^{n}\cdot n!}{n^{n}}$$converges. I know it does and I tried to use the ratio test, but in the final limit, I...
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