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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 :
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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