급수와 정적분의 관계

[출제의도] 급수와 정적분의 관계를 이용하여 값을 구할 수 있는가? $\displaystyle \lim\limits _{n \rightarrow \infty} \sum\limits _{k = 1} ^{{}_{n}} \frac{k ^{2} + 2 kn}{k ^{3} + 3 k ^{2} n + n ^{3}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} \sum\limits _{k = 1} ^{{}_{n}} \left\{ \frac{\left( \frac{k}{n} \right) ^{2} + 2 \times \frac{k}{n}}{\left( \frac{k}{n} \right) ^{3} + 3 \times \left( \frac{k}{n} \right) ^{2} + 1} \times \frac{1}{n} \right\}$ $\displaystyle = \int _{0} ^{1} \frac{x ^{2} + 2 x}{x ^{3} + 3 x ^{2} + 1} dx$ $\displaystyle = \left[ \frac{1}{3} \ln ( x ^{3} + 3 x ^{2} + 1 ) \right] _{0} ^{1}$ $\displaystyle = \frac{1}{3} ( \ln 5 - \ln 1 )$ $\displaystyle = \frac{\ln 5}{3}$