Find the radius/interval of convergence for (series posted below) . . .

Question

anonymous · Answer

$$\sum_{n=1}^{\infty} n^nx^n$$

I attempted the root test, but that told me the series was divergent . . . unless I am applying the root test incorrectly??

amistre64 · Answer

is that n^n * x^n ?

anonymous · Answer

Yes.  I'm sorry :P

amistre64 · Answer

it this tiny little laptop screen, hard to make stuff out :)

amistre64 · Answer

if i were to take a stab at this:

$$\frac{n^nx^n}{(n-1)^{n-1}x^{n-1}}$$

$$\frac{n^nx}{(n-1)^{n-1}}$$

$$|x|\lim_{n\to inf}\frac{n^n}{(n-1)^{n-1}}$$

$$|x|\lim_{n\to inf}\frac{n^n}{(n-1)^{n}(n-1)^{-1}}$$

$$|x|\lim_{n\to inf}\frac{n^n(n-1)}{(n-1)^{n}}$$

$$|x|\lim_{n\to inf}\frac{n^{n+1}-n^n}{(n-1)^{n}}$$

yeah, i got no idea where this is going yet

amistre64 · Answer

i think the top goes to zero as the bottom goes to inf

zarkon · Answer

the series only converges when x=0...I'll let you think about that :)

amistre64 · Answer

http://www.wolframalpha.com/input/?i=limit+%28n%5En%29%2F%28n-1%29%5E%28n-1%29+n+to+inf yeah, for the moment all i get is divergent

anonymous · Answer

Oh, okay, so the ratio test worked . . . and if it's 0/infinity, then . . . that's 0?  So the radius/interval would both be 0?

amistre64 · Answer

the wolf says it diverges

anonymous · Answer

Okay.  :<  Thanks to both of y'all :)

zarkon · Answer

it diverges everywhere except at x=0