How to show this converges to 0

**paupsers** · November 15th 2009, 01:29 PM

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?

**Plato** · November 15th 2009, 01:35 PM

Originally Posted by paupsers

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

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

$\frac{{2^x }}<br /> {{3^{2x + 1} }} = \frac{1}<br /> {3}\left( {\frac{2}<br /> {9}} \right)^x$

**tonio** · November 15th 2009, 01:35 PM

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

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