Question

$\lim_{n \rightarrow \infty}\sum_{k = 1}^{n}\frac{k}{n^{2} + k^{2}}$is equals to

• A. $\frac{1}{2}\log 2$
• B. $x = \frac{3\pi}{4}$
• C. $\pi/4$
• D. $\pi/2$

Answer 1

Answer: (A)

$\frac{1}{2}\log 2$

Answer 2

Let $I = \lim_{n \rightarrow \infty}\sum_{k = 1}^{n}\frac{k}{n^{2} + k^{2}}$ $= \lim_{n \rightarrow \infty}\sum_{k = 1}^{n}{}\frac{1}{n}\frac{\left( \frac{k}{n} \right)}{1 + \left( \frac{k}{n} \right)^{2}}$

$I = \int_{0}^{1}{\frac{x}{1 + x^{2}}dx}$ $= \frac{1}{2}\lbrack\log(1 + x^{2})\rbrack_{0}^{1} = \frac{1}{2}\left\lbrack \log 2 \right\rbrack$.