급수와 정적분

$S _{n}$$\displaystyle = \frac{3}{n \sqrt{n}} \sum\limits _{k=1} ^{n} \sqrt{n + 2 k}$$\displaystyle = \frac{3}{n} \sum\limits _{k=1} ^{n} \sqrt{1 + 2 \left( \frac{k}{n} \right)}$이므로 $\displaystyle \lim\limits _{n\rightarrow \infty} S _{n}$$\displaystyle = \lim\limits _{n\rightarrow \infty} \frac{3}{n} \sum\limits _{k=1} ^{n} \sqrt{1 + 2 \left( \frac{k}{n} \right)}$ $\displaystyle = \int _{0} ^{1} 3 \sqrt{1 + 2 x} dx$ $\displaystyle = \left[ ( 1 + 2 x ) ^{\frac{3}{2}} \right] _{0} ^{1}$ $\displaystyle = 3 ^{\frac{3}{2}} - 1$ ∴ $\displaystyle \lim\limits _{n\rightarrow \infty} ( S _{n} + 1 ) = \lim\limits _{n\rightarrow \infty} S _{n} + 1$$\displaystyle = 3 ^{\frac{3}{2}}$$\displaystyle = 3 \sqrt{3}$