Geometric Series
Functions & Advanced AlgebraA geometric series is the sum of the terms of a geometric sequence, with a finite or infinite number of terms.
Formula
S_n = a_1 \cdot (1 - r^n) / (1 - r)
Definition
A geometric series is the sum of the terms of a geometric sequence. You are adding numbers that each get multiplied by the same ratio.
Example
$1 + 2 + 4 + 8 + 16 = 31$ is a geometric series with ratio $2$ (finite, $5$ terms). $1 + 1/2 + 1/4 + 1/8 + \ldots$ is a geometric series with ratio $1/2$ (infinite, keeps going).
Key Insight
If the ratio is less than $1$ (like $1/2$), an infinite geometric series can add up to a finite number. Adding infinitely many smaller and smaller pieces can still have a limit.
Definition
The sum of the first $n$ terms of a geometric series: $S_n = a_1 \cdot (1 - r^n) / (1 - r)$, for $r \neq 1$. For an infinite geometric series with $|r| < 1$: $S = a_1 / (1 - r)$.
Example
Sum $3 + 6 + 12 + 24 + 48$ ($n=5$, $r=2$): $S_5 = 3 \cdot (1 - 2^5)/(1 - 2) = 3 \cdot (-31)/(-1) = 93$. Infinite: $1 + 1/3 + 1/9 + \ldots = 1/(1 - 1/3) = 1/(2/3) = 3/2$.
Key Insight
The formula $S = a/(1-r)$ for infinite geometric series is foundational. It explains repeating decimals: $0.333\ldots = 3/10 + 3/100 + \ldots = (3/10)/(1 - 1/10) = 1/3$.
Definition
The partial sum formula $S_n = a(1 - r^n)/(1 - r)$ is derived by computing $S_n - r S_n = a - ar^n$. As $n \to \infty$ with $|r| < 1$, $r^n \to 0$ and $S_n \to a/(1-r)$. The geometric series is the power series $\sum a r^n$, the simplest case of a Taylor series, converging on the disk $|r| < 1$ in $\mathbb{C}$.
Example
The power series $1/(1-x) = \sum_{n=0}^{\infty} x^n$ for $|x| < 1$ is the foundation of generating functions. Differentiating: $1/(1-x)^2 = \sum n x^{n-1}$. Integrating: $-\ln(1-x) = \sum x^n/n$.
Key Insight
The geometric series is the entry point to complex analysis: its radius of convergence, analytic continuation, and poles (at $x = 1$) are models for understanding general power series, Laurent expansions, and singularity theory.