Algebra 2 / Unit 7: Sequences and Series
7.6 Infinite Geometric Series
Notes
An infinite geometric series has a finite sum only when $|r| < 1$. Then it converges to
$$S = \frac{a_1}{1 - r}$$
- If $|r| \ge 1$, the terms don't shrink to 0 and the series diverges (no sum).
- Example: $\displaystyle 8 + 4 + 2 + 1 + \dots = \frac{8}{1 - \frac{1}{2}} = 16$.
- Repeating decimals are infinite geometric series: $\displaystyle 0.333\dots = \frac{0.3}{1 - 0.1} = \frac{1}{3}$.
Practice
Find the sum of $12 + 6 + 3 + \dots$
$\displaystyle r = \frac{1}{2}$, so $\displaystyle S = \frac{12}{1 - \frac{1}{2}} = 24$.
Which infinite series has a finite sum?
Only $r = -0.5$ has $|r| < 1$.
Write $0.777\dots$ as a fraction.
$\displaystyle \frac{0.7}{1 - 0.1} = \frac{0.7}{0.9} = \frac{7}{9}$.
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