How to test whether this series converge or not $\sum_{n=2}^{∞} \frac{1}{\left(n^5-n\right)^{\frac{1}{4}}}$? How to test whether this series converge or not $$\sum_{n=2}^{∞} \frac{1}{\left(n^5-n\right)^{\frac{1}{4}}}$$
I tired using the ratio test and that didn't work, because $\lim _{n\to \infty }\left(\frac{a_{n+1}}{a_n}\right) = 1$ which is indeterminate by the ratio test. So I also tried using the comparison test $0< a_n < b_n$ and I couldn't find a suitable $b_n$ that I am familiar with. Or do I even have to use this? Can I just use this theorem: If a series $\sum_{n=1}^{\infty}a_n$ of real numbers converges then $\lim_{n \to \infty}a_n = 0$? When do you even use the comparison? How do you tell?
Many thanks everyone.

Note that for $n\ge2$,
$$n^5-n>\frac12n^5$$
Hence
$$\sum_{n=2}^\infty\frac1{(n^5-n)^{1/4}}<\sum_{n=2}^\infty\frac1{(n^5/2)^{1/4}}=2^{1/4}\sum_{n=2}^\infty n^{-5/4}<\infty$$
