_n 1^4+2^4+3^4++n^4n^5 - _n 1^3+2^3+3^3++n^3n^5

Question

$\displaystyle\lim_{n	o\infty} \dfrac{1^4+2^4+3^4+\cdots+n^4}{n^5} - \lim_{n	o\infty} \dfrac{1^3+2^3+3^3+\cdots+n^3}{n^5}$

Accepted Answer

$= \lim_{n	o\infty}\dfrac1n\sum_{r=1}^n\left(\dfrac{r}{n}ight)^4 - \lim_{n	o\infty}\dfrac1n\cdot\lim_{n	o\infty}\dfrac1n\sum_{r=1}^n\left(\dfrac{r}{n}ight)^3$

$= \int_0^1 x^4\,dx - 0	imes\int_0^1 x^3\,dx = \left[\dfrac{x^5}{5}ight]_0^1 - 0 = \dfrac15$

Solutions · 1