Definite Integration Question 213
Question: $ \underset{n\to \infty }{\mathop{\lim }}\sum\limits _{k=1}^{n}{\frac{k}{n^{2}+k^{2}}} $ is equals to
[Roorkee 1999]
Options:
A) $ \frac{1}{2}\log 2 $
B) $ x=\frac{3\pi }{4} $
C) $ \pi /4 $
D) $ \pi /2 $
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ I=\underset{n\to \infty }{\mathop{\lim }}\sum\limits _{k=1}^{n}{\frac{k}{n^{2}+k^{2}}} $
$ =\underset{n\to \infty }{\mathop{\lim }}\sum\limits _{k=1}^{n}{{}}\frac{1}{n}\frac{( \frac{k}{n} )}{1+{{( \frac{k}{n} )}^{2}}} $
$ I=\int\limits_0^{1}{\frac{x}{1+x^{2}}dx} $
$ =\frac{1}{2}[\log (1+x^{2})] _{0}^{1} $
$ =\frac{1}{2}[ \log 2 ] $ .
BETA
