CalcRef

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).

Variables

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

Find the sum: 3 + 3/4 + 3/16 + ...
  1. a = 3, r = 1/4. Sum = 3/(1-1/4) = 3/(3/4) = 4

Answer: 4

Related formulas

Geometric Sequence

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