# How to show this converges to 0

• Nov 15th 2009, 01:29 PM
paupsers
How to show this converges to 0
Prove that
$\frac{2^x}{3^{2x+1}}$ converges to 0 as x approaches infinity.

Not quite sure how to begin this. I know that it's true, but I don't know how to prove it using definitions and theorems. Any hints?
• Nov 15th 2009, 01:35 PM
Plato
Quote:

Originally Posted by paupsers
Prove that
$\frac{2^x}{3^{2x+1}}$ converges to 0 as x approaches infinity.

$\left| r \right| < 1\, \Rightarrow \,\lim _{n \to \infty } r^n = 0$.

$\frac{{2^x }}
{{3^{2x + 1} }} = \frac{1}
{3}\left( {\frac{2}
{9}} \right)^x$
• Nov 15th 2009, 01:35 PM
tonio
Quote:

Originally Posted by paupsers
Prove that
$\frac{2^x}{3^{2x+1}}$ converges to 0 as x approaches infinity.

Not quite sure how to begin this. I know that it's true, but I don't know how to prove it using definitions and theorems. Any hints?

$\frac{2^x}{3^{2x+1}}=\frac{1}{3}\left(\frac{2}{9}\ right)^x$ , and I hope you know that if $|x|<1\,\,then\,\,x^n\xrightarrow [n\to\infty] {} 0$...

Tonio