MCQ
$\mathop {\lim }\limits_{n \to \infty } \left[ {\frac{1}{{{n^3} + 1}} + \frac{4}{{{n^3} + 1}} + \frac{9}{{{n^3} + 1}} + ........ + \frac{{{n^2}}}{{{n^3} + 1}}} \right] = $
- A$1$
- B$2/3$
- ✓$\frac{1}{3}\,$
- D$0$
$ = \mathop {\lim }\limits_{n \to \infty } \,\,\frac{1}{6}\frac{{n\,(n + 1)\,(2n + 1)}}{{1 + {n^3}}}$
$ = \mathop {\lim }\limits_{n \to \infty } \,\frac{1}{6}\frac{{\left( {1 + \frac{1}{n}} \right)\,\left( {2 + \frac{1}{n}} \right)}}{{\left( {\frac{1}{{{n^3}}} + 1} \right)}}$
$ = \frac{1}{6}\,.\,1\,.\,\frac{2}{{(1)}} = \left( {\frac{1}{3}} \right).$
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