급수를 정적분으로

$\displaystyle \lim\limits _{n \rightarrow \infty} {\sum\limits _{k = 1} ^{n} \frac{2}{n + k} f \left( 1 + \frac{k}{n} \right)}$$\displaystyle = \lim\limits _{n \rightarrow \infty} \frac{1}{n} \sum\limits _{k=1} ^{n} \frac{2}{1 + \frac{k}{n}} f \left( 1 + \frac{k}{n} \right)$ $\displaystyle = \int _{1} ^{2} \frac{2}{x} f \left( x \right) dx$ $\displaystyle = \int _{1} ^{2} 2 xe ^{x ^{2} - 1} dx$ $\cdots \cdots$ ㉠ $x ^{2} - 1 = t$라 하면 $\displaystyle 2 x = \frac{dt}{dx}$이고, $x = 1$일 때 $t = 0$, $x = 2$일 때 $t = 3$이므로 ㉠에서 $\displaystyle \int _{1} ^{2} 2 xe ^{x ^{2} - 1} dx$$\displaystyle = \int _{0} ^{3} e ^{t} dt$$\displaystyle = \left[ \begin{array}{l} e ^{t} \begin{matrix} \\ \end{matrix} \end{array} \right] _{0} ^{3}$$= e ^{3} - 1$