무한급수 수렴 조건과 극한값

$\displaystyle \sum\limits _{n = 1} ^{\infty} \left( \frac{a _{n}}{4 ^{n}} - 3 \right)$이 수렴하므로 $\displaystyle \lim\limits _{n \rightarrow \infty} {\left( \frac{a _{n}}{4 ^{n}} - 3 \right) = 0}$ $\displaystyle \left( \because \sum\limits _{n = 1} ^{\infty} a _{n} \text{이} \text{수렴} \Rightarrow \lim\limits _{n \rightarrow \infty} {a _{n} = 0} \right)$ $\displaystyle \therefore \lim\limits _{n \rightarrow \infty} {\frac{a _{n}}{4 ^{n}} = 3}$ …… ㉠ $\displaystyle \therefore \lim\limits _{n \rightarrow \infty} {\frac{a _{n} - 4 \cdot 2 ^{n}}{a _{n} - 2 \cdot 4 ^{n}}}$$\displaystyle = \lim\limits _{n \rightarrow \infty} {\frac{\frac{a _{n} - 4 \cdot 2 ^{n}}{4 ^{n}}}{\frac{a _{n} - 2 \cdot 4 ^{n}}{4 ^{n}}}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {\frac{\frac{a _{n}}{4 ^{n}} - 4 \cdot \frac{1}{2 ^{n}}}{\frac{a _{n}}{4 ^{n}} - 2}}$$\displaystyle = \frac{3 - 0}{3 - 2}$ $\displaystyle \left( \because \text{㉠} , \lim\limits _{n \rightarrow \infty} {\frac{1}{2 ^{n}} = 0} \right)$ $= 3$