Prove the following

Question:

$\lim _{n \rightarrow \infty}\left(\frac{(n+1)^{1 / 3}}{n^{4 / 3}}+\frac{(n+2)^{1 / 3}}{n^{4 / 3}}+\ldots \ldots+\frac{(2 n)^{1 / 3}}{n^{4 / 3}}\right)$ is equal to :

  1. $\frac{4}{3}(2)^{4 / 3}$

  2. $\frac{3}{4}(2)^{4 / 3}-\frac{4}{3}$

  3. $\frac{3}{4}(2)^{4 / 3}-\frac{3}{4}$

  4. $\frac{4}{3}(2)^{3 / 4}$

Correct Option: , 3

Solution:

$\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{1}{n}\left(\frac{n+r}{n}\right)^{1 / 3}$

$=\int_{0}^{1}(1+x)^{1 / 3} \mathrm{dx}=\frac{3}{4}\left(2^{4 / 3}-1\right)$

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