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  4. Evaluate: displaystyle lim n arrow ∞[(1/n3+1)+(4/n3+1)+(9/n3+1)+ ldots ldots+(n2/n3+1)]

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Q. Evaluate : $\displaystyle\lim _{n \rightarrow \infty}\left[\frac{1}{n^{3}+1}+\frac{4}{n^{3}+1}+\frac{9}{n^{3}+1}+\ldots \ldots+\frac{n^{2}}{n^{3}+1}\right]$

Limits and Derivatives

Solution:

$\displaystyle\lim _{n \rightarrow \infty}\left[\frac{1}{n^{3}+1}+\frac{4}{n^{3}+1}+\frac{9}{n^{3}+1}+\ldots . \frac{n^{2}}{n^{3}+1}\right]$
$=\displaystyle\lim _{n \rightarrow \infty} \frac{n(n+1)(2 n+1)}{6\left(n^{3}+1\right)}=\frac{1}{3}$