Limits using definite integration $F(k)$ = $$ \lim_{n\to \infty}{\frac{1^k + 2^k +...+n^k}{(1^2 + 2^2 +...+n^2)*(1^3 + 2^3 +...+n^3)}} $$
I need help in finding $F(5)$ and $F(6)$.
I tried converting it into summation form and using the progression formulas of $n^2$ and $n^3$ but it was of no use.

Theorem: Define $S_r(n)=1^r+2^2+...+n^r$, $$\frac{n^{r+1}}{r+1}\leq S_r(n)\leq\frac{(n+1)^{r+1}}{r+1}$$ for any positive $r\geq0$
Then $$\frac{12n^{k+1}}{(k+1)(n+1)^3(n+1)^4}\leq{\frac{1^k + 2^k +...+n^k}{(1^2 + 2^2 +...+n^2)\cdot(1^3 + 2^3 +...+n^3)}}\leq \frac{12(n+1)^{k+1}}{(k+1)n^3n^4}$$
So, $$F(6)=\frac{12}{7}$$ and $F(k)=0$, if  $0\leq k<6$ and $F(k)=+\infty$, if $k>6$
