you must use a series test. so far, the techniques you have hinted at using work for SEQUENCES (taking the limit of the nth term formula to see if it converges, using the Squeeze theorem etc), not series. (in fact, if you want to use limits to tell whether a series DOES NOT converge, there is the divergence test, but it is not applicable here)

the series diverges by the Limit Comparison test. Compare to the harmonic series