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: