$$\sum_{n=1}^\infty \frac{n}{\sqrt[3]{8n^5-1}}$$
From the tests that I know of:
Divergence Test: The limit is ≠ to a constant, so inconclusive.
Geometric series: I don't think this could be written in that manner.
Comparison Test/Lim Comparison: Compare to $$\frac{n}{8n^{\frac{5}{3}}}$$
Integral Test: I can't think of a integration method that would work here.
Alternating Series/Root Test don't apply.
Ratio Test: The limit is 1 so inconclusive.
Perhaps I'm making a mistake throughout the methods I've tried, but I'm lost. Using these tests, is it possible to find whether or not it converges or diverges?
Best Answer
Use comparison test,$$\frac{n}{\sqrt[3]{8n^5-1}} \ge \frac{n}{\sqrt[3]{8n^5}} = \frac{1}{2n^{\frac53-1}}=\frac{1}{2n^{\frac23}}$$
Now, use $p$-series to make conclusion that it diverges.