# Lim of P(n)/(a^n) = 0

1. Consider the sequence xn=nabn, where a is a natural number and b is a real number with 0<b<1. Show that the sequence converges to zero. Conclude from here that lim P(n)/cn=0, where P is a polynomial function and c>1.

2. I am not sure how to show that the sequence converges to zero.

3. We know that bn certainly converges to zero since 0<b<1. I have tried to show that since that part of the sequence converges to zero, the entire sequence converges to zero; however, I believe I need that at least the rest of the sequence is bounded, which it is not. I have also tried using the standard epsilon-definition of the convergence of a sequence, but that has proved to be messy, with ln's and e's. My guess is it's something simple that I'm not seeing...

Dick
Homework Helper
I would try to work with log(x_n)=log(n^a*b^n). Can you show the log goes to negative infinity as n->infinity? You can do this if you can show log(x_n)/n goes to log(b) as n->infinity.

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So taking the log's of both sides and applying rules of log's we get:

log(x_n)/n= (alogn)/n + logb. Is there a way to show (a logn)/n goes to zero as n goes to infinity without using L'Hospital's rule?

Dick