TITLE: Find the value of the expression.
QUESTION [6 upvotes]: $$ 
\sum_{i=1}^{1000} i^\frac{-2}{3} =M$$
$$\text{Then find the value of  [M]-20} $$
$$\text{where[ . ] Denotes the greatest integer function.}$$
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.

REPLY [3 votes]: $$\small\int_1^{1000}x^{-2/3}\,dx>\sum_{j=2}^{1000}(j-(j-1))\cdot j^{-2/3}\implies27>M-1\implies27<M<28\implies \lfloor M\rfloor-20=7.$$