Question

\[ \sum_{n=1}^{infty} {n!}/{100}^{n}\]

Answer 1

convergence

Answer 2

ratio test?

Answer 3

(n+1)!
------
100^(n+1)
------
n!
----
100^n

Answer 4

\[\lim_{n\to\infty}|\{a_{n+1}\}\cdot\frac1{\{a_n\}}|=\lim_{n\to\infty}|\frac{100^{n+1}}{(n+1)!}\cdot\frac{n!}{100^n}|=\lim_{n\to\infty}|\frac{100}{n+1}|=0<1\]

Answer 5

I think...

Answer 6

oh no it's upside-down!

Answer 7

yeah..but how do we get rid of the "n!"

Answer 8

\[\lim_{n\to\infty}|\{a_{n+1}\}\cdot\frac1{\{a_n\}}|=\lim_{n\to\infty}|\frac{(n+1)!}{100^{n+1}}\cdot\frac{100^n}{n!}|=\lim_{n\to\infty}|\frac{n+1}{100}|=DNE\]so divergent I guess

Answer 9

how did you solve the (n+1)! ? I don't know what to do with it

Answer 10

I just took the limit as n goes to infty\[\lim_{n\to\infty}|\{a_{n+1}\}\cdot\frac1{\{a_n\}}|=\lim_{n\to\infty}|\frac{(n+1)!}{100^{n+1}}\cdot\frac{100^n}{n!}|=\lim_{n\to\infty}|\frac{n+1}{100}|=\frac\infty{100}\]which is a non-existent limit

Answer 11

Sorry don't mean to be so stubborn. I understand that it's divergent. But here is my issue...
on a test, can I just say that lim n-> of (n+1)! = (n+1)

Answer 12

lim -> infinity of ...

Answer 13

lim n-> infinity... sorry i can't type today

Answer 14

i look at the equation and see the "!" sign and don't know what to do with it in term of limits

Answer 15

Lolz...is my question really that silly?

Answer 16

http://www.wolframalpha.com/input/?i=lim+n-%3E+inf+%28n%2B1%29!%2Fn!

Answer 17

sorry I took so long to reply

Answer 18

no worries

Answer 19

are you stuck on\[\frac{(n+1)!}{n!}=n+1\]?

Answer 20

yes

Answer 21

well look at what happens with factorial with smaller numbers...

Answer 22

\[\frac{4!}{2!}={4\cdot3\cdot\cancel2\cdot\cancel1\over\cancel2\cdot\cancel1}=4\cdot3=12\]so certain numbers cancel

Answer 23

got it...so since it's (n+1)!/n! ( n+1) will be the only term standing

Answer 24

if the number on the bottom is one less than that on the top\[\frac{6!}{5!}={6\cdot\cancel5\cdot\cancel4\cdot\cancel3\cdot\cancel2\cdot\cancel1\over\cancel5\cdot\cancel4\cdot\cancel3\cdot\cancel2\cdot\cancel1}=6\]so yeah, you got the idea

Answer 25

great! thanks!

Answer 26

welcome :)