Convergence of $\sum_{n=1}^\infty \frac{1}{(n^{2/\beta}(n^{2\beta}+1))^{1/5}}$ I'm stuck on a problem about $\sum_{n=1}^\infty \frac{1}{(n^{\frac{2}{\beta}}(n^{2\beta}+1))^{\frac{1}{5}}}$. I need to find for which values of $\beta$ this series converge (the correct answer is $0 < \beta <  \frac{1}{2}$) but no matter how I manipulate it, I don't know how to isolate beta.
I've tried Leibniz criterium and simplify it as $n^{\frac{2}{\beta}}(n^{2\beta}+1) \to \inf $   but when I do that, I find that $\beta$ must be between $ 0 < \beta < 2$ which is false obviously. I'm kind of lost, at first I thought it was because it's a polynomial but since we are looking at the limit it doesn't make sense...
I don't see any others criterium that could help so I tried to compare it to a series I know but I find it hard to put this series in a form I could find one. I think I need to isolate $\beta$ at some point, but I can't?
Thanks for reading!

Notice that :
$$\frac{1}{(n^{\frac{2}{\beta}}(n^{2\beta}+1))^{\frac{1}{5}}} \sim \frac{1}{n^{\frac{2}{5 \beta} + \frac{2\beta}{5}}}$$
We deduce that the series :
$$\sum \frac{1}{(n^{\frac{2}{\beta}}(n^{2\beta}+1))^{\frac{1}{5}}}$$
converges if, and only if, :
$$\frac{2}{5 \beta} + \frac{2\beta}{5} > 1$$
if, and only if, :
$$2 \beta^2 - 5 \beta + 2 > 0$$
if, and only if, :
$$(2 \beta - 1) (\beta - 2) > 0$$
if, and only if, :
$$\beta < \dfrac{1}{2} \text{ or } \beta > 2$$
