# How to show this converges to 0

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

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

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

Originally Posted by paupsers
Prove that
$\displaystyle \frac{2^x}{3^{2x+1}}$ 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?

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

Tonio