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

Question

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...

score 4 · Accepted Answer · answered Nov 28 '15 at 17:07

4

Since $$\lim\limits_{n\rightarrow\infty}\left(1-\frac{1}{n}\right)^n=\frac{1}{e}$$ the sum cannot converge

answered Nov 28 '15 at 17:07

Peter Woolfitt

score 3 · Answer 2 · answered Nov 28 '15 at 17:05

3

One has: $$\lim_{n\to+\infty}\left(1-\frac{1}{n}\right)^n=\frac{1}{e}\neq 0.$$ Therefore, your series does not converge.

answered Nov 28 '15 at 17:05

C. Falcon

score 0 · Answer 3 · answered Nov 28 '15 at 17:13

0

Direct comparison test: $(1-\frac{1}{n})^n<e^{-1}$, hence the divrgence

answered Nov 28 '15 at 17:13

Alex

3 Answers3