The binomial theorem is useful in determining the leading-order behavior of expressions with n negative or fractional when x is small. Of course when n is a positive integer, it reduces to the familiar expressions for polynomials with which you are familiar from your study of algebra.
where f', f'', and f(n) are derivatives with respect to x. A Maclaurin series is the special case of a Taylor series with a=0
The function (1+x)n may be expressed as a Maclaurin series by evaluating the following derivatives:
Thus the Maclaurin series for (1+x)n is the binomial theorem: