Question

By the definition of the definite integral, the value of
$\underset{n\to \infty }{lim}(\frac{1^4}{1^5+n^5}+\frac{2^4}{2^5+n^5}+\frac{3^4}{3^5+n^5}+....+\frac{n^4}{n^5+n^5})$ is

• A. log 2
• B. $\frac{1}{5}$ log 2
• C. $\frac{1}{4}$ log 2
• D. $\frac{1}{3}$ log 2

Answer 1

The correct option is B $\frac{1}{5}$ log 2

$\underset{n\to \infty }{lim}(\frac{1^4}{1^5+n^5}+\frac{2^4}{2^5+n^5}+\frac{3^4}{3^5+n^5}+....+\frac{n^4}{n^5+n^5})=\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{r^4}{r^5+n^5}$

$=\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{{\left(\frac{r}{n}\right)}^4}{1+{\left(\frac{r}{n}\right)}^5}\cdot \frac{1}{n}={\int }_0^1\frac{x^4}{1+x^5}dx=\frac{1}{5}\ln (1+x^5){|}_0^1=\frac{1}{5}\ln 2$