# Obtain exact results from GenerateConditions?

GROUPS:
 In[1]:= squareRootOfTermPlusOne[term_] := Sum[Binomial[1/2, n]*term^n, {n, 0, Infinity}, GenerateConditions -> True] In[2]:= squareRootOfTermPlusOne[x] Out[2]= ConditionalExpression[Sqrt[1 + x], Abs[x] < 1] It says the infinite sum converges if and only if Abs[x]<1 But here are counterexamples of convergence on arguments with an absolute value not less than 1: In[3]:= squareRootOfTermPlusOne /@ {+1, -1, I} Out[3]= {Sqrt[2], 0, Sqrt[1 + I]} And similarly for GeneratingFunction: In[11]:= GeneratingFunction[Binomial[1/2, n], n, x, GenerateConditions -> True] Out[11]= ConditionalExpression[Sqrt[1 + x], Abs[x] < 1] 
 Gianluca Gorni 4 Votes GenerateConditions gives sufficient conditions, apparently. The pointwise convergence on the boundary of the convergence disk is difficult to study in general.