Infinite Geometric Series Calculator

Formula

S = a/(1 − r), valid only when |r| < 1

Worked example

With First term: 3, Common ratio (|r| < 1): 0.5, infinite series sum is 6.

Calculation method

The rule is S = a/(1 − r), valid only when |r| < 1.

For this infinite geometric series example, changing first term from 3 to 4 while keeping the other inputs fixed changes infinite series sum from 6 to 8.

Reading infinite series sum

The nth term identifies one position; a series sum accumulates several positions. Fibonacci terms follow their own recurrence from earlier terms. In the worked values on this page, first term 3, common ratio (|r| < 1) 0.5 produce infinite series sum 6.

Practical use

Use these formulas for recurring patterns, repeated growth, and ordered numerical problems. The specific task here is to find infinite series sum from first term, common ratio (|r| < 1).

Limits and common mistakes

An infinite geometric series has a finite sum only when the absolute common ratio is below one. Check whether the requested index starts at zero or one. Verify first term before interpreting infinite series sum; a value measured on another basis would answer a different question.

Why the relationship works

An arithmetic pattern repeatedly adds a fixed difference, while a geometric pattern repeatedly multiplies by a fixed ratio. A single nth term and a sum of the first n terms are different results. The infinite geometric sum is a separate limiting calculation that is valid only when the ratio’s absolute value is below one. The infinite geometric series rule uses first term, common ratio (|r| < 1) in S = a/(1 − r), valid only when |r| < 1; read each symbol as the field named on this page.

Interpreting another set of values

For infinite geometric series, the fields labeled first term, common ratio (|r| < 1) are the known quantities. The calculator solves for infinite series sum using S = a/(1 − r), valid only when |r| < 1. Clarify whether the output is an integer count, a selected term, or a sum of terms. The index and the rule about order or repetition determine which mathematical object is being counted. When comparing two calculations, keep their units and input definitions identical.

For a second infinite geometric series check, keep the other default inputs fixed and change first term from 3 to 5. Infinite series sum changes from 6 to 10. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

Check whether an index starts at zero or one and whether the result is a single term or a sum. The infinite geometric sum needs an absolute common ratio below one. A positive finite series count must be a whole number. The reported infinite series sum is 6 in the worked example. Keep its stated unit or counting convention when checking it against another calculation.