Since I cannot type mathematical symbols in post, I have included an image of a series that I believe converges. In fact, I think this series converges to pi/2 - 1. I know this because it is closely related to the Taylor series for arcsin (that I have previously found).
Anyway, I was able to show that the limit of the individual terms is zero, hence the Divergence Test is inconclusive. (If you want more details on how I did that, let me know.)
I've been trying the usual tests to prove that this series converges, but I can't seem to. However, my knowledge of infinite products is small. Can anyone give me a hint?