MCQ
$\mathop {\lim }\limits_{n \to \infty } \frac{{{1^p} + {2^p} + {3^p} + ..... + {n^p}}}{{{n^{p + 1}}}} = $
- ✓$\frac{1}{{p + 1}}$
- B$\frac{1}{{1 - p}}$
- C$\frac{1}{p} - \frac{1}{{p - 1}}$
- D$\frac{1}{{p + 2}}$
$= \mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {\left[ {\frac{{{r^p}}}{{{n^{p + 1}}}}} \right]} $
$= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{r = 1}^n {{{\left( {\frac{r}{n}} \right)}^p}} = \int_0^1 {{x^p}dx} = \left[ {\frac{{{x^{p + 1}}}}{{p + 1}}} \right]_0^1 = \frac{1}{{p + 1}}$.
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$(A)$ $\mathrm{S}_{\mathrm{n}}<\frac{\pi}{3 \sqrt{3}}$ $(B)$ $S_n>\frac{\pi}{3 \sqrt{3}}$
$(C)$ $T_n<\frac{\pi}{3 \sqrt{3}}$ $(D)$ $T_n>\frac{\pi}{3 \sqrt{3}}$
$A_1=\left\{(x, y): x \geq 0, y \geq 0,2 x+2 y-x^2-y^2>1>x+y\right\}$
$A_2=\left\{(x, y): x \geq 0, y \geq 0, x+y>1>x^2+y^2\right\}$
$A_3=\left\{(x, y): x \geq 0, y \geq 0, x+y>1>x^3+y^3\right\}$
Denote by $\left|A_1\right|,\left|A_2\right|$ and $\left|A_3\right|$ the areas of the regions $A_1, A_2$ and $A_3$ respectively. Then,