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If I input:

Sum[Binomial[-1/2, n], {n, 0, ∞}]

MMA returns:

Sum::div: Sum does not converge. >>

But I think that this sum does converge. It is an alternating sum and the ratio of the absolute values of subsequent terms is less than 1. In other words, the absolute values of the terms in the sum are decreasing. It should converge between 1/2 and 1.

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How to figure out

In the documentation of Sum , under Possible Issues you can read :

Sums may not be convergent, Using Regularization may give a finite value.

enter image description here

A solution example

Sum[
 Binomial[-1/2, n]
 , {n, 0, ∞}
 , Regularization -> "Abel"
 ]
1/Sqrt[2]

Plot

ListPlot[
 Table[
  {k, NSum[Binomial[-1/2, n], {n, 0, k}]}
  , {k, 1, 1000}
  ]
 , Epilog -> {Blue, InfiniteLine[{0, 0.7071067811865475`}, {1, 0}]}
 , PlotTheme -> "Scientific"
 , PlotRange -> {0.5, 1}
 ]

Mathematica graphics

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