Thread: Limit of the real sequence nk^n

I am trying to prove the following limit in formal terms.

lim n -> inf (n x^n) = 0, where |x| < 1


I dont want to use the L'hopitals rule or any other method that involves differentiation. I am trying to use the following sequence definition.

x --> x0 =
(0 < e ==> there exists and N. such that for all n. n >= N ==>
abs (x n - x0) < e

but have got stuck in the middle.

Any hints regarding this would be appreciated.
Thanks,

Originally Posted by led5v

I am trying to prove the following limit in formal terms.

lim n -> inf (n x^n) = 0, where |x| < 1

You want to compute,
lim inf n*x^n

Instead if we can show,
lim n*x^n =0
Then the lim inf is the same.

Note,
lim n*x^n =0

Hint: Use the ratio test (for x!=0).