$$\mathop {\lim }\limits_{n \to \infty } \frac{{{1^p} + {2^p} + {3^p} + ..... + {n^p}}}{{{n^{p + 1}}}}$$      is-

Solution :
We have
$$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } \frac{{{1^p} + {2^p} + {3^p} + ..... + {n^p}}}{{{n^{p + 1}}}}; \cr & \mathop {\lim }\limits_{n \to \infty } \sum\limits_{r\, = \,1}^n {\frac{{{r^p}}}{{{n^p}.n}} = \int\limits_0^1 {{x^p}dx = \left[ {\frac{{{x^{p + 1}}}}{{p + 1}}} \right]_0^1 = \frac{1}{{p + 1}}} } \cr} $$