Why doesn’t Mathematica provide an answer while Wolfram|Alpha does, concerning a series convergence?

Among other series I’ve been working on, I was asked to find whether $ $ \sum_n 1-\cos(\frac{\pi}{n})$ $ converged, and Mathematica’s output to SumConvergence[1 - Cos[Pi/n], n] simply was repeating the input, without further information. Wolfram|Alpha, though, at least told me which test were or not conclusive.

I’m new to Mathematica, and even though I’ve looked both on Google and into Wolfram’s documentation, I haven’t found information that could help me figure out how to get, from Mathematica, the conditions for the convergence of a series involving something else than powers of a variable.

I would appreciate if you could give me some clues on the typical procedure to make Mathematica correctly evaluate the convergence of a series, or/and to return the conditions for convergence. Thank you in advance.