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).

  • 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 + ...

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solution above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

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.

Practice

Quiz yourself on Sequences & Series

Geometric Series is one of 18 Sequences & Series formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 18 Sequences & Series formulas