정적분과 급수

주어진 급수를 변형하면 $\displaystyle \lim\limits _{n \rightarrow \infty} \frac{\pi}{n} \sum\limits _{k = 1} ^{n} \cos \frac{k \pi}{4 n} = 4 \lim\limits _{n \rightarrow \infty} \frac{\pi}{4 n} \sum\limits _{k = 1} ^{n} \cos \frac{k \pi}{4 n}$ $f \left( x \right) = \cos x$, $\displaystyle x _{k} = \frac{k \pi}{4 n}$, $\displaystyle \delta x = \frac{\pi}{4 n}$라 하면 정적분과 급수의 관계에 의하여 $\displaystyle 4 \lim\limits _{n \rightarrow \infty} \frac{\pi}{4 n} \sum\limits _{k = 1} ^{n} \cos \frac{k \pi}{4 n}$$\displaystyle = 4 \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{n} f \left( x _{k} \right) \Delta x$ $\displaystyle = 4 \int _{0} ^{\frac{\pi}{4}} \cos x dx$ $\displaystyle = 4 \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} \sin x \end{array} \right] _{0} ^{\frac{\pi}{4}}$ $\displaystyle = 4 \left( \sin \frac{\pi}{4} - \sin 0 \right)$ $\displaystyle = 4 \times \frac{\sqrt{2}}{2}$$\displaystyle = 2 \sqrt{2}$