S=\frac{a_1}{1-r},\quad |r|<1Variables
- Define every symbol and unit before substitution.
- Check the domain, shape, and convention required by the formula.
How to use this formula
Computes the sum of a convergent infinite geometric series.
Important notes
- Verify assumptions and units before applying the expression.
- Keep exact values until the final rounding step when possible.
Quick example
Use the infinite geometric series sum with a small known example, then verify the result independently.
Applicability, worked calculation, and verification
Domain and applicability
Apply Infinite Geometric Series Sum only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- For the Infinite Geometric Series Sum, verify assumptions and units before applying the expression.
- For the Infinite Geometric Series Sum, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For the Infinite Geometric Series Sum, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.
Do not use this formula when
- Do not use Infinite Geometric Series Sum when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Infinite Geometric Series Sum, check zero, negative, and extreme input values before relying on the result.
- When using Infinite Geometric Series Sum, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form S=\frac{a_1}{1-r},\quad |r|<1 for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For 3+1.5+0.75+…, a₁=3 and r=1/2, so S=3/(1-1/2)=6.
- Verify |r|=1/2<1.
- Compute 1-r=1/2.
- Divide 3 by 1/2 to obtain 6.
Independent verification
Partial sums 3, 4.5, 5.25, 5.625 approach 6 from below.
Common mistakes
- When copying Infinite Geometric Series Sum, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- For the Infinite Geometric Series Sum, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Infinite Geometric Series Sum in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
S=\frac{a_1}{1-r},\quad |r|<1. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Infinite Geometric Series Sum.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Infinite Geometric Series Sum used for?
Computes the sum of a convergent infinite geometric series.
Can I copy this formula as LaTeX?
Yes. Copy S=\frac{a_1}{1-r},\quad |r|<1 or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.