What is the value of $$[(1^ {-2/3}) + (2^ {-2/3}) + (3^ {-2/3}) + … + (1000^ {-2/3})]?$$ where $[x]$ stands for the greatest integer function.
P. S : I ran a code on my PC to find that the quantity inside the [ ] function is approximately equal to $27.5$ , thus $[27.5] = 27$. I am however Looking for a mathematical way to approximate the sum, instead of having run a code.