Geometric Series

A basic geometric series is, perhaps, a familiar example: we start with a value a₀ and then obtain each new term of the series by multiplying by a common ratio. In keeping with our focus on functions, let us use x for the common ratio and examine the convergence of

$$f(x) = \sum_{n=0}^\infty a_0 x^n$$

The k^th partial sum of this geometric series is

$$S_k(x) =\sum_{n=0}^k a_0 x^k.$$

We can find its value through the following calculation:

$$\begin{eqnarray*}S_k(x) &=&\sum_{n=0}^k a_0 x^n \\ x S_k(x) &=& \sum_{n=1}^{k+1}... ...=& a_0 - a_0 x^{k+1} \\ S_k(x) &=& \frac{ a_0 - a_0 x^{k+1}}{1-x}\end{eqnarray*}$$

Now this final expression will converge to the limit

$$f(x) = \frac{a_0}{1-x}$$

when $x^{k+1}\to 0$. This in turn happens if and only if |x|<1. In summary,

$$\frac{a_0}{1-x} = \sum_{n=0}^\infty a_0 x^n \mbox{ provided }\vert x\vert<1.$$

Examples: Several functions can be given series expansions by decorating this result:

• $${\frac{1}{1+x}}$$ is geometric with a₀= 1 common ratio -x and series representation $${ \sum_{n=0}^\infty (-1)^nx^n}$$ valid for $\vert x\vert< 1 \\$.
• $${\frac{x}{1-x}}$$ is geometric with a₀= x common ratio x and series representation $${ \sum_{n=1}^\infty x^n}$$ valid for |x|< 1.
• $${\frac{2}{1+3x}}$$ is geometric with a₀= 2 common ratio -3 x and series representation $${ \sum_{n=0}^\infty 2\ (-3)^n x^n}$$ valid for $\vert x\vert< \frac13$.
• $${\frac{1}{2-x}}$$ is geometric with $a_0= \frac12$ common ratio $\frac{x}2$ and series representation $${ \sum_{n=0}^\infty \frac{1}{2^{n+1}}x^n}$$ valid for |x|< 2.
• $${\frac{1}{1-x^2}}$$ is geometric with a₀= 1 common ratio x² and series representation $${ \sum_{n=0}^\infty x^{2n}}$$ valid for |x|< 1.

Exercises: As an exercise to see if you have the technique try to find a series representation of the following functions, giving the values for x for which the series converges.

1.

$${\frac{1}{1+x^2}}$$

2.

$${\frac{x}{1-x^3}}$$

3.

$${\frac{3}{4+2x^2}}$$

4.

$${\frac{x^2}{1+4x}}$$

5.

$${\frac{3x}{5-x^5}}$$

6.

$${\frac{x+1}{1+x^2}}$$

7.

$${\frac{3}{3+2x^2}}$$