Thread: ratio test problem

sum n=1 oo n^n / (n)!

a_n+1 / a_n =

edit: looked funny when i typed it in...
(n+1)^(n+1) / (n+1)! times n! / n^n

=

[n^n(n+1)^n(n+1)] / n+1 = n^n(n+1)^n

did i do the algebra right? what do i do next? ><

Originally Posted by jeph

sum n=1 oo n^n / (n)!

Ratio test.

(n+1)^(n+1)/(n+1)! * n!/n^n

(n+1)^(n+1)/[(n+1)*n!] * n!/n^n

(n+1)^n/n^n

[(n+1)/n]^n

[1+1/n]^n

The limit is e>1

Thus it does not converge.

Originally Posted by ThePerfectHacker

Ratio test.
[(n+1)/n]^n

[1+1/n]^n

The limit is e>1

Thus it does not converge.

how did you turn the first n to a 1 in the first part?

how did you get e? i dont get it...

Originally Posted by jeph

how did you turn the first n to a 1 in the first part?

how did you get e? i dont get it...

(n+1)/n = (n/n)+(1/n)=1+1/n

Now the limit you need to memorize is that,

(1+1/n)^n ---> e