교점 x좌표와 극한

$a _{m}$은 두 곡선 $\displaystyle y = \frac{2 \pi}{x}$와 $y = \cos x$의 교점의 $x$좌표이므로 $\displaystyle \frac{2 \pi}{a _{m}} = \cos \left( a _{m} \right)$ $\displaystyle n \times \cos ^{2} \left( a _{n + k} \right) = n \times \frac{4 \pi ^{2}}{\left( a _{n + k} \right) ^{2}}$ $a _{1} = 2 \pi$, $m > 1$에서 $m \pi < a _{m} < \left( m + 1 \right) \pi$이므로 $\displaystyle \frac{4 n}{\left( n + k + 1 \right) ^{2}} < n \times \cos ^{2} \left( a _{n + k} \right) < \frac{4 n}{\left( n + k \right) ^{2}}$이다. $\displaystyle \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{n} \frac{4 n}{\left( n + k \right) ^{2}}$$\displaystyle = \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{n} \frac{4}{\left( 1 + \frac{k}{n} \right) ^{2}} \times \frac{1}{n}$ $\displaystyle = \int _{0} ^{1} {} \frac{4}{\left( 1 + x \right) ^{2}} dx = \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} - \frac{4}{1 + x} \end{array} \right] _{0} ^{1} = 2$ 이고 $\displaystyle \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{n} \left\{ \frac{4 n}{\left( n + k + 1 \right) ^{2}} - \frac{4 n}{\left( n + k \right) ^{2}} \right\}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \left\{ \frac{4 n}{\left( 2 n + 1 \right) ^{2}} - \frac{4 n}{\left( n + 1 \right) ^{2}} \right\} = 0$이므로 $\displaystyle \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{n} \frac{4 n}{\left( n + k + 1 \right) ^{2}} = 2$ 수열의 극한의 대소 관계의 성질에 의하여 $\displaystyle \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{n} \left\{ n \times \cos ^{2} \left( a _{n + k} \right) \right\} = 2$