cos nπ 포함 수열 극한

모든 자연수 $n$에 대하여 $\cos n \pi = ( - 1 ) ^{n}$이므로 $\displaystyle \lim\limits _{n \rightarrow \infty} {} \frac{n ^{2} + n \cos n \pi}{n ^{2} + 1} = \lim\limits _{n \rightarrow \infty} {} \frac{1 + \frac{1}{n} \cos n \pi}{1 + \frac{1}{n ^{2}}}$$\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{1 + \frac{( - 1 ) ^{n}}{n}}{1 + \frac{1}{n ^{2}}}$$= 1$ [다른 풀이] 모든 자연수 $n$에 대하여 $- 1 \leq \cos n \pi \leq 1$이므로 $\displaystyle - \frac{n}{n ^{2} + 1} \leq \frac{n}{n ^{2} + 1} \cos n \pi \leq \frac{n}{n ^{2} + 1}$ 그런데 $\displaystyle \lim\limits _{n \rightarrow \infty} {} \frac{n}{n ^{2} + 1} = \lim\limits _{n \rightarrow \infty} {} \frac{\frac{1}{n}}{1 + \frac{1}{n ^{2}}} = 0$이므로 $\displaystyle \lim\limits _{n \rightarrow \infty} {} \frac{n}{n ^{2} + 1} \cos n \pi = 0$ 또, $\displaystyle \lim\limits _{n \rightarrow \infty} {} \frac{n ^{2}}{n ^{2} + 1} = \lim\limits _{n \rightarrow \infty} {} \frac{1}{1 + \frac{1}{n ^{2}}} = 1$ ∴ $\displaystyle \lim\limits _{n \rightarrow \infty} {} \frac{n ( n + \cos n \pi )}{n ^{2} + 1}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \frac{n ^{2}}{n ^{2} + 1} + \lim\limits _{n \rightarrow \infty} {} \frac{n}{n ^{2} + 1} \cos n \pi$ $= 1$