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Sequences & Series · Series Types

Geometric Series

∑n=0∞arn=a1−rif ∣r∣<1\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} \quad \text{if } |r| < 1

A geometric series converges if and only if |r| < 1, and its sum is a/(1-r).

  • 2 variables
  • 2 worked examples
  • 18 in Sequences & Series

Variables

Variables used in the Geometric Series formula
SymbolNameUnit
aFirst term -
rCommon ratio -

Worked examples

Find the sum: Σ (1/2)ⁿ from n=0 to ∞.
  1. a = 1, r = 1/2. Sum = 1/(1-1/2) = 2

Answer: 2

1 more worked examplePremium

Find the sum: 3 + 3/4 + 3/16 + ...

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Related formulas

Geometric Sequence

See all 18 Sequences & Series formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Geometric Series?

A geometric series converges if and only if |r| < 1, and its sum is a/(1-r). It is one of the sequences & series formulas in the CalcRef reference.

What do the symbols in the Geometric Series mean?

In the Geometric Series, a is the first term; r is the common ratio.

How do you use the Geometric Series?

Worked example. Find the sum: Σ (1/2)ⁿ from n=0 to ∞. a = 1, r = 1/2. Sum = 1/(1-1/2) = 2. Answer: 2.

Practice

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