Find the integral part of the series.
$1/4^{1/3}$$+$$1/5^{1/3}$$+$$1/6^{1/3}$$...$$+$$1/1000000^{1/3}$.
I tried to bring it down to a summation form, hence reducing it to
$$\sum_{i=4}^n 1/i^{1/3}$$
But i still do not understand how to make it short. I tried a lot to use telescoping strategy, but i failed. Also, the "integral part" worries me.
The cube root of $4$ must lie between $1$ and $2$. Similarly, the cube root of $5$ must lie between $1$ and $2$ again. This occurs until the $8$ number, whose cube root is $2$. But later, i found out it would be difficult for me if i use the strategy of breaking it down. Next, i do not see anything else. Please help me!
Edit: In our competitions we are not supposed to use calculus approaches, so please avoid calculus while answering this question