PHYS208 Fundamentals of Physics II

Binomial theorem

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

[Taylor series]

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

[Maclaurin series]

The function (1+x)n may be expressed as a Maclaurin series by evaluating the following derivatives:

[Derivation]

Thus the Maclaurin series for (1+x)n is the binomial theorem:

[Binomial theorem]

Application of the Binomial Theorem


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Last updated March 10, 1998.
Copyright George Watson, Univ. of Delaware, 1997.