I want to find out for which $p \in \mathbb{R}_{>0}$ the integral
$\int_{[0,1]^n} \frac{\mathrm d x}{(x_1^p+2x_2^p + ... + nx_n^p)^{1/3}}$
converges. To be honest, I have no idea or whatsoever how to do this. I computed the integral for $n=1$ and $n=2$ and found them to be existing for $p < 3n$ so far. But Mathematica takes forever to compute the integral for $n=3$, so I think I have to find a smart way to verify whether my guess is correct.
Can anyone give me a little hint, so that I can proceed? The problem is that we've never done such things before during classes or in exercises, so I really don't know where to begin.
Thanks in advance.