Geometric Series
A 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, numbers each multiplied by the same ratio, and it can be finite or infinite. The sum of the first $n$ terms is $S_n = a_1 \cdot (1 - r^n) / (1 - r)$ for $r \neq 1$, and for an infinite geometric series with $|r| < 1$, $S = a_1 / (1 - r)$; this partial-sum formula is derived by computing $S_n - r S_n = a - ar^n$, and as $n \to \infty$ with $|r| < 1$, $r^n \to 0$ so $S_n \to a/(1-r)$. The geometric series is in fact the power series $\sum a r^n$, the simplest case of a Taylor series, converging on the disk $|r| < 1$ in $\mathbb{C}$.
Example
$1 + 2 + 4 + 8 + 16 = 31$ is a finite geometric series with ratio $2$, while $1 + 1/2 + 1/4 + 1/8 + \ldots$ is infinite with ratio $1/2$; summing $3 + 6 + 12 + 24 + 48$ gives $S_5 = 3 \cdot (1 - 2^5)/(1 - 2) = 93$, and the infinite series $1 + 1/3 + 1/9 + \ldots$ sums to $1/(1 - 1/3) = 3/2$. This formula also explains repeating decimals: $0.333\ldots = (3/10)/(1 - 1/10) = 1/3$. The power series $1/(1-x) = \sum_{n=0}^{\infty} x^n$ for $|x| < 1$ is the foundation of generating functions; differentiating gives $1/(1-x)^2 = \sum n x^{n-1}$, and integrating gives $-\ln(1-x) = \sum x^n/n$.
Key Insight
If the ratio is less than $1$, an infinite geometric series can add up to a finite number: adding infinitely many smaller and smaller pieces can still have a limit, which is the foundational fact behind the formula $S = a/(1-r)$. 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.