Wolphram alpha does not recognize the binomial coefficient as a converging series. Test: summation - Wolfram|Alpha
(1) They all give finite results. That’s about as explicitly convergent as it gets.
(2) It’s very difficult to understand what you are doing from these URLs. Presumably there was a query on the word “summation” followed by setting some example parameters. It would be more clear to readers if you were to state explicitly what you did (and how the result disagreed with what you expected).
Will could you please edit your post and just type Wolfram|Alpha queries as text?
that’s odd, it works now