Ratio test.

1. Oct 26, 2007

azatkgz

Suppose that $$a_n\geq 0$$ and there is

$$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}=c$$
If c>1,series diverges.
if c<1 series converges.

For $$a_n=\frac{n!}{n^n}$$

$$\lim_{n\rightarrow\infty}\frac{(n+1)!/(n+1)^{n+1}}{n!/n^n}$$

$$\lim_{n\rightarrow\infty}\frac{n^n}{(n+1)^n}$$

Then I used I'Hopital Rule and got answer 1.

Last edited: Oct 26, 2007
2. Oct 26, 2007

Dick

I think you'd better check your l'Hopital. Your final limit is closely related to the limit of (1+1/n)^n, which is a famous limit and is not one.

Last edited: Oct 26, 2007
3. Oct 26, 2007

azatkgz

A,yes I get it.

$$\lim_{n\rightarrow\infty}\frac{1}{(1+\frac{1}{n})^n}=\frac{1}{e}$$

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