구분구적법으로 계산하는 급수의 극한

$\displaystyle \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{2 n} \frac{k ^{2} ( 5 k ^{2} + 3 )}{n ^{3} ( n ^{2} + 1 )}$ $\displaystyle = \lim\limits _{n\rightarrow \infty} \sum\limits _{k=1} ^{2n} \frac{5 k ^{4} + 3 k ^{2}}{n ^{3} ( n ^{2} + 1 )}$ $\displaystyle = \lim\limits _{n\rightarrow \infty} \left\{ \frac{n ^{2}}{n ^{2} + 1} \cdot \sum\limits _{k=1} ^{2n} 5 \left( \frac{k}{n} \right) ^{4} \cdot \frac{1}{n} + \frac{1}{n ^{2} + 1} \cdot \sum\limits _{k=1} ^{2n} 3 \left( \frac{k}{n} \right) ^{2} \cdot \frac{1}{n} \right\}$ $\displaystyle = \lim\limits _{n\rightarrow \infty} \frac{n ^{2}}{n ^{2} + 1} \cdot \lim\limits _{n\rightarrow \infty} \sum\limits _{k=1} ^{2n} 5 \left( \frac{k}{n} \right) ^{4} \cdot \frac{1}{n}$$\displaystyle + \lim\limits _{n\rightarrow \infty} \frac{1}{n ^{2} + 1} \cdot \lim\limits _{n\rightarrow \infty} \sum\limits _{k=1} ^{2n} 3 \left( \frac{k}{n} \right) ^{2} \cdot \frac{1}{n}$ $\displaystyle = 1 \cdot \int _{0} ^{2} {} 5 x ^{4} dx + 0 \cdot \int _{0} ^{2} {} 3 x ^{2} dx$ $\displaystyle = \left[ \begin{array}{l} x ^{5} \begin{matrix} \\ \end{matrix} \end{array} \right] _{0} ^{2}$$= 2 ^{5} = 32$