But that won't be sufficient, because for large n,
diverges, but the series in the original post consists of smaller terms, so you can't use that as a direct comparison. You can however, use the fact that
and the fact that the below diverges:
In fact the 2 in the above series can be ANY number greater than 1.
Yes, the (ordinary) comparison test is if where is a convergent series, then converges. Or if where is a divergent series, then diverges.
For the limit comparison test, if is finite and nonzero, then converges if and only if converges.
So you need to compare to 1/n, since is finite and nonzero. Of course, 1/2n or 35/87n would also work - the limits would still be finite and nonzero.