[HERE][1] is an example in MO.  For the first question, I could evaluate the series in closed form, so I could compute with their known properties, and thus answer the question.  For the second question, the series were just ${}_2F_1$ series, and I did not know their properties, so I could not answer the question.  Despite the numerical evidence in the graphs.


  [1]: https://mathoverflow.net/questions/99166/convergence-of-newton-series-for-sin-ax/99176#99176