converges to zero (I think)

D'Alembert's ratio test, or simply the ratio test, is a way of determining whether certain series converge. It goes like this: to check if a series converges, check the sequence of ratios between consecutive terms. If that sequence converges to something less than 1, then the series converges absolutely. If it converges to something greater than 1, or diverges, then the series diverges. If it converges to 1 exactly, then the test is inconclusive.

When you take the integral using the series as integrand, it converges if the integral worked out to be a number. If it's infinte, the series diverge.

The definition of the word converges is to tend to meet in a point or line; incline towards each other, as lines that are not parallel or to tend to a common result or conclusion.

It converges.