$$ \sum_{i=1}^{1000} i^\frac{-2}{3} =M$$
$$\text{Then find the value of [M]-20} $$ $$\text{where[ . ] Denotes the greatest integer function.}$$ My attempt, I was able to figure out the lower end of M as $$\sum_{i=1}^{1000} i^\frac{-2}{3} \gt \int_{1}^{1000} x^\frac{-2}{3} dx$$ which yielded the result that $$ M \gt 27$$ But now I struggle to setup an upper bound on this thing. Any help is appreciated.
$$\small\int_1^{1000}x^{-2/3}\,dx>\sum_{j=2}^{1000}(j-(j-1))\cdot j^{-2/3}\implies27>M-1\implies27<M<28\implies \lfloor M\rfloor-20=7.$$