정적분과 급수

함수 $f \left( x \right) = e ^{x ^{2}}$의 양변을 $x$에 대하여 미분하면 $f' \left( x \right)$$= 2 x e ^{x ^{2}}$$= 2 x f \left( x \right)$ $\displaystyle \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{n} \frac{k}{n ^{2}} f \left( \frac{k}{n} \right)$에서 $\displaystyle \frac{k}{n} = x _{k}$라 하면 $\displaystyle \Delta x = \frac{1}{n}$이다. $\therefore$ $\displaystyle \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{n} \frac{k}{n ^{2}} f \left( \frac{k}{n} \right)$$\displaystyle = \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{n} \frac{k}{n} f \left( \frac{k}{n} \right) \frac{1}{n}$ $\displaystyle = \int _{0} ^{1} {} x f \left( x \right) dx$ $\displaystyle = \frac{1}{2} \int _{0} ^{1} {f' \left( x \right)} dx$ $\displaystyle = \frac{1}{2} \left[ \begin{array}{l} \begin{matrix} \\ \end{matrix} f \left( x \right) \end{array} \right] _{0} ^{1}$ $\displaystyle = \frac{1}{2} \left\{ f \left( 1 \right) - f \left( 0 \right) \right\}$ $\displaystyle = \frac{1}{2} e - \frac{1}{2}$