탄젠트 교점 수열의 극한

[출제의도] 수열의 극한의 성질을 이용하여 극한값을 구할 수 있는가? (ⅰ) $a _{1} + \pi < a _{2}$ $a _{2} + \pi < a _{3}$ $a _{3} + \pi < a _{4}$ $\vdots$ $a _{n - 1} + \pi < a _{n}$ 위의 식을 변끼리 더하면 $( a _{1} + a _{2} + \cdots + a _{n - 1} ) + ( n - 1 ) \pi < ( a _{2} + a _{3} + \cdots + a _{n} )$ $a _{1} + ( n - 1 ) \pi < a _{n}$ $\displaystyle \frac{\pi}{4} + ( n - 1 ) \pi < a _{n}$ $\displaystyle \frac{\pi}{4 n} + \frac{n - 1}{n} \pi < \frac{a _{n}}{n}$ $\displaystyle \lim\limits _{n \rightarrow \infty} {\left( \frac{\pi}{4 n} + \frac{n - 1}{n} \pi \right) \leq \lim\limits _{n \rightarrow \infty} {\frac{a _{n}}{n}}}$ $\displaystyle \therefore \pi \leq \lim\limits _{n \rightarrow \infty} {\frac{a _{n}}{n}}$ (ⅱ) $\displaystyle a _{n} < ( n - 1 ) \pi + \frac{\pi}{2} \text{이므로}$ $\displaystyle a _{n} < \frac{2 n - 1}{2} \pi , \frac{a _{n}}{n} < \frac{2 n - 1}{2 n} \pi$ $\displaystyle \lim\limits _{n \rightarrow \infty} {} \frac{a _{n}}{n} < \lim\limits _{n \rightarrow \infty} {} \frac{2 n - 1}{2 n} \pi$ $\displaystyle \therefore \lim\limits _{n \rightarrow \infty} {\frac{a _{n}}{n}} \leq \pi$ (ⅰ), (ⅱ)에서 $\displaystyle \pi \leq \lim\limits _{n \rightarrow \infty} {\frac{a _{n}}{n} \leq \pi \text{이므로} \lim\limits _{n \rightarrow \infty} {\frac{a _{n}}{n}} = \pi}$