정적분과 급수

$\displaystyle \lim\limits _{n \rightarrow \infty} \frac{1}{n} \sum\limits _{k = 1} ^{n} f \left( \frac{n + k}{n} \right)$$\displaystyle = \lim\limits _{n \rightarrow \infty} \sum\limits _{k = 1} ^{n} f \left( 1 + \frac{k}{n} \right) \frac{1}{n}$ $\displaystyle = \int _{1} ^{2} f ( x ) dx$ $\displaystyle = \int _{1} ^{2} \left( \frac{x + 1}{x ^{2}} \right) dx$ $\displaystyle = \int _{1} ^{2} \left( \frac{1}{x} + \frac{1}{x ^{2}} \right) dx$ $\displaystyle = \left[ \ln \left| x \right| - \frac{1}{x} \right] _{1} ^{2}$ $\displaystyle = \frac{1}{2} + \ln 2$