The only tests my class has learned so far and is allowed to use are: Divergence, Integral, Comparison, Geometric/Harmonic/Telescopic. I have proved that the series converges, via a ratio test, but my teacher doesn't want me to use that. I don't think I can do an integral test; it defnitely isn't a geometric series because the numerator increases arithmetically. It isn't any sort of p-series, not telescopic. That leaves the comparison test -- except I don't know what to compare it to. I tried $b_n = 1/3^n$, and while this converges, it's smaller than $a_n$.
As a side note, are there any tricks for finding good comparisons, or do you just develop an intuition over time?