직사각형 넓이 급수

[출제의도] 무한급수의 합을 정적분을 활용하여 구할 수 있는가를 묻는 문제이다. $\mathrm{A} ( \mathit{n} + k , n + k + 1 )$, $\mathrm{B} ( \mathit{n} + k , 0 )$ $\displaystyle \mathrm{\overline{AB}} = \mathit{n} + k + 1$이므로 $f ( k ) = 2 ( n + k + 1 )$ $\displaystyle \lim\limits _{n\rightarrow \infty} \sum\limits _{k=0} ^{n-1} \frac{1}{f ( k )} = \lim\limits _{n\rightarrow \infty} \sum\limits _{k=0} ^{n-1} \frac{1}{2 ( n + k + 1 )}$ $\displaystyle = \frac{1}{2} \lim\limits _{n\rightarrow \infty} \sum\limits _{k=1} ^{n} \frac{1}{n + k}$ $\displaystyle = \frac{1}{2} \lim\limits _{n\rightarrow \infty} \sum\limits _{k=1} ^{n} \left( \frac{1}{1 + \frac{k}{n}} \right) \frac{1}{n}$ $\displaystyle = \frac{1}{2} \int _{0} ^{1} \frac{1}{x + 1} dx = \left[ \ln ( x + 1 ) \right] _{0} ^{1} = \frac{\ln 2}{2}$