geometric series

IPA/dʒˌiːəʊmˈɛtɹɪk sˈiəɹiz/
IPA/dʒˌiːoʊmˈɛtɹɪk sˈɪɹiz/

geometric series — noun

1. a sequence of numbers where each term after the first is the previous term multi

1.nounB2
Definition

a sequence of numbers where each term after the first is the previous term multiplied by a fixed number (the ratio), and the terms are written as a sum; for example, 5 + 10 + 20 + 40 is a geometric series with ratio 2, while 27 + 9 + 3 + 1 has ratio 1/3.

Example

Renata modelled a bouncing ball using a geometric series with a fixed rebound ratio.

geometric series with a fixed [ratio]

Jiwoo expressed 0.8888… as the geometric series 0.8 + 0.08 + 0.008 + …, which equals 8/9.

repeating decimal expressed as a geometric series

Common collocations
  • convergent geometric series
  • divergent geometric series
  • infinite geometric series
  • finite geometric series
  • sum of a geometric series
  • common ratio of a geometric series
Antonyms
  • arithmetic series

    a sum of terms with a constant difference rather than a constant ratio

Grammar Patterns

a geometric series + of + [number]

Usage Note

Distinguish from an arithmetic series, where the difference between consecutive terms is constant (e.g., 5 + 10 + 15 + 20). In a geometric series the ratio between terms is constant — check whether you are adding a fixed number (arithmetic) or multiplying by a fixed number (geometric).

Common Mistakes

The sequence 5, 10, 15, 20 forms a geometric series.
The sequence 5, 10, 15, 20 forms an arithmetic series.
In a geometric series each term is multiplied by a fixed ratio, but here the difference between consecutive terms is a constant 5, making it an arithmetic series.

Synonym discussion

Sense: noun/1

A geometric series and a geometric progression (or geometric sequence) are closely related: the progression is the ordered list of numbers (e.g., 3, 6, 12, 24), while the series is the sum of those terms (3 + 6 + 12 + 24). An arithmetic series differs in that its terms change by a constant addition or subtraction rather than by multiplication — for example, 3 + 6 + 9 + 12 is arithmetic, not geometric. A convergent geometric series is a special case where the ratio lies between -1 and 1, causing the sum of infinitely many terms to approach a finite number. A power series, like x + x² + x³ + …, is a type of geometric series only when each term's coefficient follows the same ratio; more generally a power series can have any coefficients and need not form a geometric pattern.

geometric series ↗geometric progression ↗arithmetic seriesconvergent seriespower series

Etymology

GreekLatinEnglish

The word 'geometric' traces back to the Greek 'geōmetria' (gē 'earth' + metria 'measurement'), reflecting geometry's origins in land measurement. 'Series' comes from Latin 'series' meaning 'a row, chain, or sequence'. The mathematical phrase 'geometric series' arose in the 16th–17th centuries to describe sequences whose ratios, rather than differences, are constant — a concept that early geometers studied in the context of proportional segments.

First known use

1909

1909, in the meaning defined above