부등식 조건 극한

[출제의도] 수열의 극한의 성질을 이용하여 극한값을 구한다. $1 + 2 + 2 ^{2} + \cdots + 2 ^{n - 1} < a _{n} < 2 ^{n}$에서 $\displaystyle \frac{1 \times ( 2 ^{n} - 1 )}{2 - 1} < a _{n} < 2 ^{n}$ $2 ^{n} - 1 < a _{n} < 2 ^{n}$ $\displaystyle 1 - \frac{1}{2 ^{n}} < \frac{a _{n}}{2 ^{n}} < 1$ $\displaystyle \lim\limits _{n\rightarrow \infty} \left( 1 - \frac{1}{2 ^{n}} \right) = 1$, $\displaystyle \lim\limits _{n\rightarrow \infty} 1 = 1$이므로 $\displaystyle \frac{\lim\limits _{n\rightarrow \infty {a _{n}}}}{2 ^{n}} = 1$ $\displaystyle \frac{3 n - 1}{n + 1} < \sum\limits _{k=1} ^{n} b _{k} < \frac{3 n + 1}{n}$에서 $\displaystyle \lim\limits _{n\rightarrow \infty} \frac{3 n - 1}{n + 1} = 3$, $\displaystyle \lim\limits _{n\rightarrow \infty} \frac{3 n + 1}{n} = 3$이므로 $\displaystyle \lim\limits _{n\rightarrow \infty} \sum\limits _{k=1} ^{n} b _{k} = 3$ ∴ $\displaystyle \lim\limits _{n\rightarrow \infty} b _{n} = 0$ $\displaystyle \lim\limits _{n \rightarrow \infty} {} \frac{8 ^{n} - 1}{4 ^{n - 1} a _{n} + 8 ^{n + 1} b _{n}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{1 - \frac{1}{8 ^{n}}}{\frac{1}{4} \times \frac{a _{n}}{2 ^{n}} + 8 \times b _{n}}$ $\displaystyle = \frac{1 - 0}{\frac{1}{4} \times 1 + 8 \times 0}$$= 4$