등비수열 극한과 급수

$\left\{ a _{n} \right\}$이 등비수열이므로 $a _{n} = a \times r ^{n}$으로 놓으면 $\displaystyle \lim\limits _{n \rightarrow \infty} {\frac{3 ^{n}}{a _{n} + 2 ^{n}} = \lim\limits _{n \rightarrow \infty} {\frac{3 ^{n}}{a \times r ^{n} + 2 ^{n}}}}$$\displaystyle = \lim\limits _{n \rightarrow \infty} {\frac{1}{a \times \left( \frac{r}{3} \right) ^{n} + \left( \frac{2}{3} \right) ^{n}}}$ $\displaystyle \lim\limits _{n\rightarrow \infty} {\left( \frac{2}{3} \right) ^{n}} = 0$이므로 $\displaystyle \lim\limits _{n\rightarrow \infty} {a \times \left( \frac{r}{3} \right) ^{n}} = \frac{1}{6}$이어야 한다. 즉, $\displaystyle a = \frac{1}{6}$, $r = 3$ 따라서 $\displaystyle a _{n} = \frac{1}{6} \times 3 ^{n}$이므로 $\displaystyle \sum\limits _{n = 1} ^{\infty} \frac{1}{a _{n}}$$\displaystyle = 6 \sum\limits _{n = 1} ^{\infty} \left( \frac{1}{3} \right) ^{n}$$\displaystyle = \frac{2}{1 - \frac{1}{3}}$$= 3$