# Thread: [SOLVED] prove inf {|s_n|} = 0

1. ## [SOLVED] prove inf {|s_n|} = 0

Basically, I need to show that

$\displaystyle \lim_{n\to\infty}s_n=0$, where

$\displaystyle s_n=nk^n(1-k^2)$ with $\displaystyle |k|>1$.

I have already shown that $\displaystyle |s_n|$ is decreasing for all $\displaystyle n\geq N$, where

$\displaystyle N>\frac{|k|}{1-|k|}$

Thus I need only show that $\displaystyle \inf\{|s_n|\}=0$. The problem is, I'm at a loss to do so.

Any ideas?

2. Originally Posted by hatsoff
Basically, I need to show that

$\displaystyle \lim_{n\to\infty}s_n=0$, where

$\displaystyle s_n=nk^n(1-k^2)$ with $\displaystyle |k|>1$.

I have already shown that $\displaystyle |s_n|$ is decreasing for all $\displaystyle n\geq N$, where

$\displaystyle N>\frac{|k|}{1-|k|}$

Thus I need only show that $\displaystyle \inf\{|s_n|\}=0$. The problem is, I'm at a loss to do so.

Any ideas?
Are you sure for |k|>1 $\displaystyle |s_{n}|$ is decreasing ??

3. Originally Posted by xalk
Are you sure for |k|>1 $\displaystyle |s_{n}|$ is decreasing ??
Hey, you're right! I forgot to flip the inequality when I multiplied by $\displaystyle (1-|k|)$.

Thanks!