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.


1 Answer 1


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

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


  {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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.