Question

Series

Answer 1

\[\sum_{n=1}^{\infty} \frac{ n! }{ n^n }\]

Answer 2

@ganeshie8

Answer 3

I did ratio test and got 1, but isn't that inconclusive then, but if I use ratio test and harmonic series I got divergent.

But it says with ratio test it's convergent, mhm?

Answer 4

ratio test is giving me the limit 1/e

Answer 5

1/e < 1 so the series converges

Answer 6

check your work again

Answer 7

ill brb in 5 min and ill show my steps, sorry!

Answer 8

k

Answer 9

\[\huge \lim_{n \rightarrow \infty} \left| \frac{ \frac{ (n+1)! }{ (n+1)^{n+1} } }{ \frac{ n! }{ n^n } } \right|\]
\[\huge \lim_{n \rightarrow \infty} \left| \frac{ (n+1)(n)! }{ (n+1)^n(n+1) } \times \frac{ n^n }{ n! } \right|\]

Answer 10

\[\huge \lim_{n \rightarrow \infty} \left| \frac{ n^n }{ (n+1)^n } \right| \implies \lim_{n \rightarrow \infty} \left( \frac{ n }{ (n+1) } \right)^n\]

Answer 11

\[\huge \lim_{n \rightarrow \infty} \left( \frac{ n/n }{ n/n+1/n } \right)^{n/n} \implies 1\]

this is what I did or am I suppose to do the y = (x/(x+1))^x business?

Answer 12

how on earth are you allowed to divide the exponent by n ?

Answer 13

Lol I had a feeling I wasn't suppose to do that

Answer 14

So it's an indeterminate form right?

Answer 15

I can see where this is going now and how to get 1/e

Answer 16

Don't judge me! I know that was a rookie mistake, I was being lazy!

Answer 17

if you want an indeterminate form :

\[ \large \lim_{n \rightarrow \infty} \left( \frac{ n }{ n+1 } \right)^n = e^{\lim\limits_{n \rightarrow \infty} n \ln \left( \frac{ n }{ n+1} \right) }\]

Answer 18

Yeah haha I see that

Answer 19

however you can just use the standard limit (1+1/n)^n = e

Answer 20

I was trying to slip through without doing much work

Answer 21

Yeah every time I see something like that I think of e, I knew it had to do something with it, but I instead embarrassed myself

Answer 22

let wolfram do the limits

Answer 23

sup

Answer 24

Alright lol, thanks :D Now lets not tell anyone about this.

Answer 25

http://www.wolframalpha.com/input/?i=+%5Clim_%7Bn+%5Crightarrow+%5Cinfty%7D+%5Cleft%28+%5Cfrac%7B+n+%7D%7B+n%2B1+%7D+%5Cright%29%5En++

Answer 26

This one is pretty tricky if you flip the question, you'll get e and then divergent, I guess I was thinking about that instead

Answer 27

the site is acting up

Answer 28

i know i can text or receive texts

Answer 29

Yeah, I keep losing connection and what not. They need to really fix the site, or reboot if that's possible haha.