Sum of an Infinite Geometric Series

To examine the sum of all the terms of an infinite geometric sequence, we need to consider $S_n = \dfrac{u_1(1-r^n)}{1-r}$ when $n$ gets very large.
Sum to Infinity

Sum to Infinity

If $\left|r\right|>1$, the series is said to be divergent and the sum infinitely large.
For instance, when $r=2$ and $u_1=1$;
$S_\infty=1+2+4+8+\cdots$ is infinitely large.

If $\left|r\right|<1$, or $-1 \lt r \lt 1$, then as $n$ becomes very large, $r^n$ approaches $0$.
For instance, when $r=\dfrac{1}{2}$ and $u_1=1$;
$S_\infty=1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\cdots = 2$.

$$S_\infty=\dfrac{u_1}{1-r}$$ We call this the limiting sum of the series.

This result can be used to find the value of recurring decimals.
Let's take a look at $0.\overline{7}$ to see how to convert it to a fraction.
\( \begin{align} \displaystyle 0.\overline{7} &= 0.7 + 0.07 + 0.007 + 0.0007 + \cdots \\ &= 0.7 + 0.7(0.1) + 0.7(0.1)^2 + 0.7(0.1)^3 + \cdots \\ &= \dfrac{0.7}{1-0.1} \\ &= \dfrac{0.7}{0.9} \\ &= \dfrac{7}{9} \\ \end{align} \)

Example 1

Write $0.\overline{12}$ as a rational number.

Example 2

Write $0.1 \overline{2}$ as a rational number.