# Convergence of series $\sum_{n=2}^\infty\left (1-\frac 1n\right)^n$

I would like to know whether the following series $$\sum_{n=2}^\infty \left(1-\frac 1n\right)^n$$ converges.

The root test and ratio test are inconclusive. And I can't apply the Weierstrass M-test...

Since $$\lim\limits_{n\rightarrow\infty}\left(1-\frac{1}{n}\right)^n=\frac{1}{e}$$ the sum cannot converge
One has: $$\lim_{n\to+\infty}\left(1-\frac{1}{n}\right)^n=\frac{1}{e}\neq 0.$$ Therefore, your series does not converge.
Direct comparison test: $(1-\frac{1}{n})^n<e^{-1}$, hence the divrgence