How to use the binomial series to expand a function

The binomial series lets us expand expressions like a binomial raised to a negative or fractional power. The first step is to get the inside into the form one plus something. Then we can read the coefficients and check where the expansion works.

Start with the generalized binomial series

\begin{array}{l} (1 + u)^{k} = 1 + k u \\ + \frac{k (k - 1)}{2!} u^{2} \\ + \frac{k (k - 1) (k - 2)}{3!} u^{3} + \dots \end{array}

The coefficient of $u^{n}$ comes from multiplying $n$ successive factors starting with $k$ , then dividing by $n!$ . The expansion is valid for $| u | < 1$ . When $k$ is a nonnegative integer, the expansion terminates and becomes a polynomial.

Expand one over the square of one minus x

For $(1 - x)^{- 2}$ , use $k = - 2$ and $u = - x$ . The coefficient before substitution is $(- 1)^{n} (n + 1)$ because the numerator is $(- 2) (- 3) \dots (- n - 1)$ . The second factor of $(- 1)^{n}$ comes from $(- x)^{n}$ .

\begin{array}{l} (1 - x)^{- 2} = 1 + 2 x + 3 x^{2} + 4 x^{3} + \dots \\ = \sum_{n = 0}^{\infty} (n + 1) x^{n} \end{array}

Test the radius and both endpoints

The absolute ratio tends to $| x |$ , so the radius is $1$ . At either endpoint, the terms have magnitude $n + 1$ and therefore never approach zero. The interval of convergence is $(- 1, 1)$ .

Learn mathKrista KingFebruary 25, 2019math, learn online, online course, online math, equilibrium solutions

How to use the binomial series to expand a function

Start with the generalized binomial series

Expand one over the square of one minus x

Test the radius and both endpoints

Courses