It has the form,
It must be finite because if is a positive integer then eventually among we hit zero, and the series shall terminate.
But if you think of as a non-integer then it will go on forever. And it will be the binomial (infinite) expansion. Which will converge to the function.
The general form used here is:
Note if is an integer the coefficients are all zero for .
You also have to worry about convergence for non-integer powers.