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

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

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

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

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

stuck

Question

$$\sum_{n=1}^{\infty} \frac{n!}{n^{n}}$$
=
$$\lim_{ n \rightarrow \infty} \frac{(n+1)!}{(n+1)^{n+1}} \frac{n^{n}}{n!}$$
=

$$\lim_{ n \rightarrow \infty} \frac{n^{n}}{(n+1)^{n+1}} \frac{(n+1)!}{n!}$$
=

$$\lim_{ n \rightarrow \infty} \frac{n^{n}}{(n+1)^{n+1}} (n+1)$$
=

$$\lim_{ n \rightarrow \infty} \frac{n^{n}}{(n+1)^{n}} $$
=

$$\lim_{ n \rightarrow \infty} \frac{1}{(1+ \frac{1}{n})^{n}} $$
=

stuck

turingtest · Answer

exponentiate?

zarkon · Answer

look at the definition of e

zarkon · Answer

at least one of the definitions of e

anonymous · Answer

interesting

turingtest · Answer

@Zarkon  ok, I hate doing it that way, but I should get used to different methods
I have seen you, among others, use that way before though, and I do see it

anonymous · Answer

1/e

zarkon · Answer

yes

anonymous · Answer

(1/e) < 1
absolute convergence

zarkon · Answer

$$\frac{n!}{n^{n}}=\frac{n}{n}\frac{n-1}{n}\cdots\frac{3}{n}\frac{2}{n}\frac{1}{n}$$

$$\le\frac{2}{n}\frac{1}{n}=\frac{2}{n^2}$$

since $$\frac{n}{n}\frac{n-1}{n}\cdots\frac{3}{n}<1$$

zarkon · Answer

I like the comparison test for this one

nenadmatematika · Answer

MathSofiya you're solution is correct...but are you sure that the right way to say is absolute convergence...''absolute'' is mainly used in solving alternating series...just to be sure...maybe in your country you use it anyways...:D

anonymous · Answer

lolz...according to stewart's calculus  definity of the "ratio test"...

anonymous · Answer

hope it's credible...

nenadmatematika · Answer

OK,just checking because here in Europe we say convergent or divergent for series which are not alternating...but if converges absolutely then converges..so nevermind :D

anonymous · Answer

I see. Just out of curiosity...Where in Europe?

nenadmatematika · Answer

Serbia

anonymous · Answer

Very Cool!

anonymous · Answer

More power to Open Study!!! Math students from all over the globe...I just think that's soo cool!!

turingtest · Answer

OpenStudy is made up of people from all over the world. I personally know someone online from each continent besides Antarctica :)

anonymous · Answer

Amazing!