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]