Convergence of Factorial Series: Investigating the Radius of Convergence

[lmannoia]

Homework Statement

The radius of convergence of the sum (from n=1 to infinity) of n!x^n /n^n

Homework Equations

The Attempt at a Solution

I ask way too many calculus questions on here..
This is everything I've done, written really badly..

Ratio test:
(n+1)!xⁿ⁺¹/(n+1)ⁿ⁺¹ times nⁿ/(n!)xⁿ
Once I simplify it all, I get down to nⁿ/(n+1)ⁿ
Does the limit approach infinity then? Did I make a mistake doing the ratio test?

 

[micromass]

You are right that the ratio test yields

[tex]\lim_{n\rightarrow +\infty}{\frac{n^n}{(n+1)^n}}[/tex]

So you need to calculate this limit. To do this, write [tex](n+1)^n=n^n(1+1/n)^n[/tex]. You should see a famous limit popping up...

 

[lmannoia]

So the limit is 1/e, making the radius of convergence e then?

 

[micromass]

Yes, I believe that is correct.

 

[icystrike]

You are right that the ratio test yields

[tex]\lim_{n\rightarrow +\infty}{\frac{n^n}{(n+1)^n}}[/tex]

So you need to calculate this limit. To do this, write [tex](n+1)^n=n^n(1+1/n)^n[/tex]. You should see a famous limit popping up...

hmm.. i think he've missed out his [tex]x[/tex] didnt he?

 

[micromass]

I never write the x I guess you could do it either way...