A geometric series is a series of the form $\sum_{n=0}^\infty ar^n = a + ar + ar^2 + ar^3 + \cdots$, where $a$ is the first term and $r$ is the common ratio. Geometric series are the simplest infinite series — they converge if and only if $|r| < 1$, and their sum can be found using a simple formula.
🎯 In this section you will learn
- The form of a geometric series and how to identify $a$ and $r$
- The convergence condition: $|r| < 1$
- The sum formula: $\sum_{n=0}^\infty ar^n = \frac{a}{1-r}$
- How to find the sum of geometric series starting at $n=1$ or other indices
- Repeating decimals as geometric series
📌 The Geometric Series Formula
$$ \sum_{n=0}^\infty ar^n = a + ar + ar^2 + \cdots = \frac{a}{1-r} \quad \text{if } |r| < 1 $$
$$ \text{If } |r| \ge 1, \text{ the series diverges} $$
💡 Derivation
Partial sum: $S_N = a + ar + ar^2 + \cdots + ar^{N-1} = a\frac{1-r^N}{1-r}$.
As $N \to \infty$, $r^N \to 0$ if $|r| < 1$, so $S = \frac{a}{1-r}$.
Find the sum of $\sum_{n=0}^\infty \frac{1}{2^n} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$.
- $a = 1$, $r = \frac{1}{2}$, $|r| = 0.5 < 1$ → converges.
- Sum $= \frac{1}{1 - 1/2} = \frac{1}{1/2} = 2$.
Find the sum of $\sum_{n=1}^\infty \frac{1}{3^n} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots$.
①
Identify $a$ and $r$
First term $a = \frac{1}{3}$, $r = \frac{1}{3}$
②
Apply formula
Sum $= \frac{1/3}{1 - 1/3} = \frac{1/3}{2/3} = \frac{1}{2}$
Find the sum of $\sum_{n=0}^\infty \left(-\frac{1}{2}\right)^n = 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \cdots$.
- $a = 1$, $r = -\frac{1}{2}$, $|r| = 0.5 < 1$ → converges.
- Sum $= \frac{1}{1 - (-1/2)} = \frac{1}{1 + 1/2} = \frac{1}{3/2} = \frac{2}{3}$.
Show that $\sum_{n=0}^\infty 2^n = 1 + 2 + 4 + 8 + \cdots$ diverges.
- $a = 1$, $r = 2$, $|r| = 2 \ge 1$ → diverges.
- Partial sums grow without bound: $S_N = 2^N - 1 \to \infty$.
Write $0.\overline{7} = 0.7777\ldots$ as a geometric series and find its sum.
①
Write as a series
$0.7777\ldots = \frac{7}{10} + \frac{7}{100} + \frac{7}{1000} + \cdots = \sum_{n=1}^\infty \frac{7}{10^n}$
②
Identify $a$ and $r$
$a = \frac{7}{10}$, $r = \frac{1}{10}$
③
Find the sum
Sum $= \frac{7/10}{1 - 1/10} = \frac{7/10}{9/10} = \frac{7}{9}$
For what values of $x$ does $\sum_{n=0}^\infty x^n$ converge? Find the sum.
- This is a geometric series with $a=1$, $r=x$.
- Converges when $|x| < 1$, sum $= \frac{1}{1-x}$.
- Diverges when $|x| \ge 1$.
⚠️ Common Mistakes
- Mistaking $a$: The first term depends on the starting index. For $\sum_{n=1}^\infty ar^n$, $a$ is the first term (when $n=1$).
- Forgetting the condition $|r| < 1$: Geometric series only converge when the absolute value of the ratio is less than 1.
- Sign errors with negative $r$: $1 - r$ in the denominator becomes $1 - (negative) = 1 + |r|$.
🔍 Key Takeaways
- Geometric series: $\sum_{n=0}^\infty ar^n$ converges iff $|r| < 1$.
- Sum: $\frac{a}{1-r}$ (when starting at $n=0$). For other starting indices, adjust accordingly.
- Repeating decimals are geometric series.
- The geometric series is the most important series to recognize — it appears in many contexts.
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