급수를 정적분으로

$\displaystyle \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = n + 1} ^{2 n} \frac{\sqrt{k}}{n \sqrt{n}}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \left( \sqrt{1 + \frac{1}{n}} + \sqrt{1 + \frac{2}{n}} + \sqrt{1 + \frac{3}{n}} + \cdots \sqrt{1 + \frac{n}{n}} \right) \frac{1}{n}$ $\displaystyle = \lim\limits _{n \rightarrow \infty} {} \sum\limits _{k = 1} ^{n} \sqrt{1 + \frac{k}{n}} \cdot \frac{1}{n}$$\displaystyle = \int _{0} ^{1} {\sqrt{1 + x}} dx$ $\displaystyle = \int _{1} ^{2} {\sqrt{x}} dx$ $\therefore a + b = 3$