The binomial theorem for integer exponents can be generalized to fractional exponents. The associated Maclaurin series give rise to some interesting identities (including generating functions) and other applications in calculus.
For example, is not a polynomial. While positive integer powers of can be expanded into polynomials e.g. this does not make a polynomial, so there cannot be a finite sum of monomial terms that equals . But there is a way to recover the same type of expansion if infinite sums are allowed.
As a first approximation, since , the tangent line to at is . So for small This approximation is already quite useful, but it is possible to approximate this function more carefully using series.
Expand as a Maclaurin series.
Now, let Then
So the Maclaurin series equals
It can be shown by the ratio test that the series converges for .
The above example generalizes immediately for all fractional exponents . Let be a real number and a positive integer. Define
Then the same analysis as in the example gives
Let be a rational number ( integers). Then
which converges for .
The binomial coefficients can sometimes be rewritten in interesting ways.
Let be a positive integer. Then
Alternatively, this can be written with a double factorial simply as
for . Substituting for gives the result that the generating function for the central binomial coefficients is