Obtain exact results from GenerateConditions?

Questions
1

In[1]:= squareRootOfTermPlusOne[term_] :=
Sum[Binomial[1/2, n]*term^n, {n, 0, Infinity},
GenerateConditions ā†’ 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]
2

GenerateConditions gives sufficient conditions, apparently. The pointwise convergence on the boundary of the convergence disk is difficult to study in general.