I am trying to investigate the convergence of the series:
\begin{equation} \sum_{n=1}^{\infty} \bigg ( 1 -\frac{1}{\sqrt{n}} \bigg )^n. \end{equation}
According to the Mathematica, the series converges, but I am unable to come up with a series with which I can compare the series above etc. Any leads would be appreciated.
Since $ \left(1-\frac{1}{\sqrt{n}}\right)^{\sqrt{n}}\underset{n\to +\infty}{\longrightarrow}\mathrm{e}^{-1} $, we have that $ \left(1-\frac{1}{\sqrt{n}}\right)^{n}\underset{n\to +\infty}{\sim}\mathrm{e}^{-\sqrt{n}} $
And since $ \mathrm{e}^{-\sqrt{n}}=\underset{\overset{n\to +\infty}{}}{o}\left(\frac{1}{n^{2}}\right) $, we get that $ \sum\limits_{n\geq 1}{\mathrm{e}^{-\sqrt{n}}} $ converges, which means $ \sum\limits_{n\geq 1}{\left(1-\frac{1}{\sqrt{n}}\right)^{n}} $ also converges.
