- #1

- 3

- 0

_{n}=n

^{a}b

^{n}, 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)/c

^{n}=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 b

^{n}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...