# How do you determine whether the sequence a_n=(2^n+3^n)/(2^n-3^n) converges, if so how do you find the limit?

Mar 29, 2017

${\lim}_{n \to \infty} {a}_{n} = - 1$

#### Explanation:

Divide both numerator and denominator by ${3}^{n}$ like this...

${\lim}_{n \to \infty} {a}_{n} = {\lim}_{n \to \infty} \frac{{2}^{n} + {3}^{n}}{{2}^{n} - {3}^{n}}$

$\textcolor{w h i t e}{{\lim}_{n \to \infty} {a}_{n}} = {\lim}_{n \to \infty} \frac{{\left(\frac{2}{3}\right)}^{n} + 1}{{\left(\frac{2}{3}\right)}^{n} - 1}$

$\textcolor{w h i t e}{{\lim}_{n \to \infty} {a}_{n}} = \frac{0 + 1}{0 - 1}$

$\textcolor{w h i t e}{{\lim}_{n \to \infty} {a}_{n}} = - 1$