Question

The value of $\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{r^3}{r^4+n^4}$ is

• A. $\frac{1}{2}{\log }_e\frac{1}{2}$
• B. $\frac{1}{4}{\log }_e\frac{1}{2}$
• C. $\frac{1}{4}{\log }_{e}2$
• D. $\frac{1}{2}{\log }_{e}2$

Answer 1

Answer: (C)

$\frac{1}{4}{\log }_{e}2$

$\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{r^3}{r^4+n^4}$

$\underset{n\to \infty }{lim}\sum \frac{n^3{\left(\frac{r}{n}\right)}^3}{n^4[{\left(\frac{r}{n}\right)}^4+1]}$

$={\int }_0^1\frac{x^3}{1+x^4}dx={\left[\frac{1}{4}{\log }_e(1+x^4)\right]}_0^1$

$=\frac{1}{4}{\log }_e\left(2\right)-0=\frac{1}{4}{\log }_e\left(2\right)$