Definite Integration Question 372
Question: $ \underset{n\to \infty }{\mathop{\lim }}\frac{1+2^{4}+3^{4}+….+n^{4}}{n^{5}} $
$ -\underset{n\to \infty }{\mathop{\lim }}\frac{1+2^{3}+3^{3}+….+n^{3}}{n^{5}}= $
[AIEEE 2003]
Options:
A) $ \frac{1}{30} $
B) Zero
C) $ \frac{1}{4} $
D) $ \frac{1}{5} $
Show Answer
Answer:
Correct Answer: D
Solution:
$ \underset{n\to \infty }{\mathop{\lim }}\frac{1}{n}{ {{( \frac{1}{n} )}^{4}}+{{( \frac{2}{n} )}^{4}}+{{( \frac{3}{n} )}^{4}}+…+{{( \frac{n}{n} )}^{4}} }- $
$ $
$ \underset{n\to \infty }{\mathop{\lim }}\frac{1}{n}{ ( \frac{1}{n^{4}} )+( \frac{2^{3}}{n^{4}} )+……+( \frac{n^{3}}{n^{4}} ) } $
$ =\int_0^{1}{{{(x)}^{4}}dx-0=[ \frac{x^{5}}{5} ]_0^{1}=\frac{1}{5}.} $
BETA
