Definite Integration Question 208
Question: $ \underset{n\to \infty }{\mathop{\lim }}\frac{1}{n}\sum\limits _{r=1}^{2n}{\frac{r}{\sqrt{n^{2}+r^{2}}}} $ equals
[IIT 1997 Re-exam]
Options:
A) $ 1+\sqrt{5} $
B) $ -1+\sqrt{5} $
C) $ -1+\sqrt{2} $
D) $ 1+\sqrt{2} $
Show Answer
Answer:
Correct Answer: B
Solution:
$ L=\underset{n\to \infty }{\mathop{\lim }}\sum\limits _{r=1}^{2n}{{}}\frac{1}{n}.\frac{r/n}{\sqrt{1+{{(r/n)}^{2}}}} $ = $ \int_0^{2}{{}}\frac{x}{\sqrt{1+x^{2}}}dx=\sqrt{5}-1 $ .
BETA
