2
$\begingroup$

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...

$\endgroup$
4
$\begingroup$

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

$\endgroup$
3
$\begingroup$

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

$\endgroup$
0
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.