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  • <img alt="\lim _{n \rightarrow \infty}\left[\frac{1}{1+n^3}+\frac{4}{8+n^3}+\frac{9}{27+n^3}+\ldots .+\frac{1}{2 n}\right]=" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bn%20%5

\lim _{n \rightarrow \infty}\left[\frac{1}{1+n^3}+\frac{4}{8+n^3}+\frac{9}{27+n^3}+\ldots .+\frac{1}{2 n}\right]=

Option: 1

\frac{1}{3} \log 2


Option: 2

\log 2


Option: 3

0


Option: 4

None of these


Answers (1)

best_answer

\lim _{n \rightarrow \infty}\left[\frac{1}{1+n^{3}}+\frac{4}{2^{3}+n^{3}}+\ldots \ldots \ldots+\frac{n^{2}}{n^{3}+n^{3}}\right]=\lim _{n \rightarrow \infty}\left[\frac{r^{2}}{r^{3}+n^{3}}\right]=\lim _{n \rightarrow \infty} \frac{1}{n}\left[\frac{\frac{r^{2}}{n^{2}}}{\left(1+\frac{r^{3}}{n^{3}}\right)}\right]=\int_0^1 \frac{x^2}{1+x^3} d x   let 1+x^3=t \\

 3 x^2 d x=d t \Rightarrow           x^2dx=\frac {dt}{3}      

Hence (A) is the correct answer.

Posted by

sudhir.kumar

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