Geometric Series of Real Numbers

# Geometric Series of Real Numbers

One very useful and important type of series is known as a geometric series. Perhaps these type of series are the "easiest" to work with. We define this class of series below.

 Definition: A Geometric Series is a series of the form $\displaystyle{\sum_{n=1}^{\infty} ax^{n-1} = a + ax + ax^2 + ...}$ where $a \in \mathbb{R}$.

For example, the following is an example of a geometric series:

(1)
\begin{align} \quad \sum_{n=1}^{\infty} x^{n-1} = 2 + 2x + 2x^2 + ... \end{align}

Whether such a geometric series converges or diverges depends only on the value of $x$. As we will see in the following theorem, if $\mid x \mid < 1$ then $\displaystyle{\sum_{n=1}^{\infty} ax^{n-1}}$ will converge and the sum will actually very nice.

 Theorem 1: If $\mid x \mid < 1$ then the geometric series $\displaystyle{\sum_{n=1}^{\infty} ax^{n-1}}$ converges and $\displaystyle{\sum_{n=1}^{\infty} ax^{n-1} = \frac{a}{1 - x}}$ and if $\mid x \mid \geq 1$ then $\displaystyle{\sum_{n=1}^{\infty} ax^{n-1}}$ diverges.
• Proof: Consider the sequence of partial sums $(s_n)_{n=1}^{\infty}$ for this series whose general term is given as:
(2)
\begin{align} \quad s_n = \sum_{k=1}^{n} x^k \end{align}
• Multiply both sides by $(1 - x)$ to get:
(3)
\begin{align} \quad (1 - x)s_n &= (1 - x) \sum_{k=1}^{n} ax^{k-1} \\ \quad (1 - x)s_n &= \sum_{k=1}^{n} a(1 - x)x^{k-1} \\ \quad (1 - x)s_n &= \sum_{k=1}^{n} a[x^{k-1} - x^{k}] \\ \quad (1 - x)s_n &= a[1 - x] + a[x - x^2] + a[x^2 - x^3] + ... + a[x^{n-1} - x^n] \\ \quad (1 - x)s_n &= a[1 - x^n] \\ \quad s_n &= \frac{a[1 - x^n]}{1 - x} \\ \end{align}
• Suppose that $\mid k \mid < 1$. Then we see that:
(4)
\begin{align} \quad \lim_{n \to \infty} s_n = \lim_{n \to \infty} \frac{a[1 - x^n]}{1 - x} = \frac{a}{1 - x} \end{align}
• Therefore we have that $\displaystyle{\sum_{k=1}^{\infty} ax^{n-1} = \frac{a}{1 - x}}$.