MCQ
$\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{1}^{99}}+{{2}^{99}}+{{3}^{99}}+......{{n}^{99}}}{{{n}^{100}}}=$
- A$\frac{9}{{100}}$
- ✓$ 1/100$
- C$\frac{1}{{99}}$
- D$\frac{1}{{101}}$
$= \mathop {\lim }\limits_{n \to \infty } \,\sum\limits_{r = 1}^n {\,\left( {\frac{{{r^{99}}}}{{{n^{100}}}}} \right)} $
$ = \mathop {\lim }\limits_{n \to \infty } \,\frac{1}{n}\,\,\sum\limits_{r = 1}^n {\,{{\left( {\frac{r}{n}} \right)}^{99}} = \int_0^1 {{x^{99}}dx = \left[ {\frac{{{x^{100}}}}{{100}}} \right]_0^1 = \frac{1}{{100}}.} } $
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