nice-calculus-simple-limit-lim-n-1-a-1-2-a-1-n-a-1-n-1-a-2-a-n-a-where-a-2-1-

Question Number 124738 by mnjuly1970 last updated on 05/Dec/20

\dots n i c e c a l c u l u s \dots

s i m p l e l i m i t ::

{l i m}_{n \to \infty} {\frac{1^{a + 1} + 2^{a + 1} + \dots + n^{a + 1}}{n (1^{a} + 2^{a} + \dots . n^{a})}} = ?

w h e r e a \neq - 2, - 1

Commented by PRITHWISH SEN 2 last updated on 07/Dec/20

\lim_{n \to \infty} \frac{\underset{r = 1}{\sum^{n}} {(\frac{r}{n})}^{a + 1}}{\underset{k = 1}{\sum^{n}} {(\frac{k}{n})}^{a}} = \frac{\int_{0}^{1} x^{a + 1} dx}{\int_{0}^{1} x^{a} dx} = \frac{a + 1}{a + 2}

Commented by mnjuly1970 last updated on 14/Dec/20

t h a n k y o u s i r . .

Commented by PRITHWISH SEN 2 last updated on 14/Dec/20

welcome