Question

$\displaystyle\lim_{n\to\infty} \left(\frac{\left(n+1\right)^{\frac{1}{3}} }{n^{\frac{4}{3}}} + \frac{\left(n+2\right)^{\frac{1}{3}}}{n^{\frac{4}{3}}} + ..... + \frac{\left(2n\right)^{\frac{1}{3}}}{n^{\frac{4}{3}}}\right) $ equal to :

• A. $\frac{4}{3} \left(2\right)^{\frac{4}{3}} $
• B. $\frac{3}{4} \left(2\right)^{\frac{4}{3}} - \frac{4}{3} $
• C. $\frac{3}{4} \left(2\right)^{\frac{4}{3}} - \frac{3}{4} $
• D. $\frac{4}{3} \left(2\right)^{\frac{3}{4}} $

Answer 1

Answer: (C)

$\lim_{n\to\infty} \sum^{n}_{r=1} \frac{1}{n} \left(\frac{n+r}{n}\right)^{1/3} $
$ = \int^{1}_{0} \left(1+x\right)^{1/3} dx = \frac{3}{4} \left(2^{4/3} -1\right) $