Geometric Series¶

Proposition¶

Let $(a_i)_{i=1}^n$ be a finite geometric sequence defined by $a_i=aq^{i-1}$ where $a,q\in\mathbb{R}$ and $q$ is known as the common ratio. The (finite) geometric series formula is given by

\sum_{i=1}^n\left(aq^{i-1}\right)=\begin{cases} \displaystyle \,\frac{a\left(1-q^n\right)}{1-q},& \text{if } q\neq 1,\\[2pt] \displaystyle \,an,&\text{if } q= 1. \end{cases}

Taking the limit of the finite partial sums gives the infinite geometric series formula

\sum_{i=1}^\infty aq^{i-1}=\frac{a}{1-q} \qquad\text{when }|q|<1.

If $a\neq0\text{,}$ the series diverges when $|q|\geq1\text{.}$ If $a=0\text{,}$ every term is zero and the series converges for every $q\text{.}$

Interpretation¶

A geometric series models repeated proportional change: each term is obtained by multiplying the previous one by the fixed ratio $q$ . It is one of the few infinite series whose convergence behaviour and sum can be described explicitly, which makes it a standard benchmark for other series.

A unit square repeatedly halved into rectangles of area 1/2, 1/4, 1/8, 1/16 and so on, which together fill the whole square, so the series sums to 1.

The square has area $1$ . The pieces $\tfrac{1}{2},\tfrac{1}{4},\tfrac{1}{8},\ldots$ form the geometric series with first term $a=\tfrac{1}{2}$ and ratio $q=\tfrac{1}{2}$ $\frac{1}{2}$ . After $n$ pieces the uncovered part has area $(\tfrac{1}{2})^n$ , so the partial sums $1-(\tfrac{1}{2})^n$ tend to $1$ , in agreement with $a/(1-q)=1$ $1$ .

With $|q|<1$ the partial sums settle at $\frac{a}{1-q}$ ; with negative $q$ they approach it from alternate sides. Push $|q|$ to $1$ or beyond and they stop settling.

For a finite series, subtracting $q$ times the sum from the sum cancels the intermediate terms. This gives $(1-q)S_n=a(1-q^n)$ $q^{n})$ ; the finite calculation comes before the limiting argument for an infinite series. Everything then depends on $q^n$ . When $|q|<1$ $1$ , $q^n\to0$ and the sum is $a/(1-q)$ $q)$ . When $|q|\ge1$ and $a\neq0$ $0$ , the terms do not tend to $0$ and the series diverges; for $q=-1$ the partial sums $a,0,a,0,\ldots$ oscillate.

A common slip concerns the first term. The formula $a/(1-q)$ is for a series whose first term is $a$ : $\displaystyle \sum_{i=1}^\infty q^i=q/(1-q)$ $q)$ , not $1/(1-q)$ $q)$ , because its first term is $q$ . For $q=\tfrac{1}{2}$ this gives $1$ , against $2$ for the series in the Examples. Geometric sums return later, for example in the probabilities of the geometric distribution and in the ratio test of Math III.

Key Results¶

If $a\neq0\text{,}$ the infinite geometric series converges if and only if $|q|<1\text{.}$ The all-zero series is the only exception.

Examples¶

$\displaystyle\sum_{i=1}^4 2^{i-1}=15\text{.}$
$\displaystyle\sum_{i=1}^\infty \left(\frac{1}{2}\right)^{i-1}=2\text{.}$

Prerequisites¶

Arithmetic Series

Sources¶

Math I: Calculus · Lecture 10

Aliases: geometric sum Keywords: geometric progression, common ratio Symbols: element of, infinity, real numbers, sum

Last updated: 20 October 2026 03:02