## PHYS208 Fundamentals of Physics II

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.

### Derivation:

You may derive the binomial theorem as a Maclaurin series. Recall that a
Taylor series relates a function *f(x)* to its value at any arbitrary
point *x=a* by

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:

"http://www.physics.udel.edu/~watson/phys208/binomial.html"

Last updated Sept. 21, 1997.

Copyright George Watson, Univ. of Delaware, 1997.