Infinite Geometric Series Sum Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The sum of a convergent infinite geometric series (with −1 < r < 1) is calculated with S = a ÷ (1 − r), where a is the first term and r the common ratio. For a = 1 and r = 0.5 (the series 1 + 0.5 + 0.25 + 0.125 + ...), the sum is exactly 2.
Explanation
A geometric series adds up indefinitely the terms of a geometric sequence — each term obtained by multiplying the previous one by a constant ratio r. What seems paradoxical at first (adding an infinite number of terms) nonetheless gives a finite, perfectly defined result, provided the ratio stays strictly between −1 and 1: each term then becomes smaller and smaller, so the sum approaches a precise limit value without ever exceeding it. The case a = 1, r = 0.5 is a classic example known as Zeno's paradox (or the dichotomy paradox): covering half a distance, then half of what remains, then half again, and so on to infinity, amounts to covering exactly the total distance despite an infinite number of steps — the sum 1 + 0.5 + 0.25 + 0.125 + ... is exactly 2, not infinity. If instead |r| ≥ 1, each term stays the same size or grows indefinitely, and the total sum diverges (it grows without limit, or oscillates without ever settling) — which is why this calculator bounds the ratio strictly between −1 and 1. This behavior contrasts sharply with that of an arithmetic sequence: an infinite arithmetic series (where each term is obtained by ADDING the same constant value, rather than multiplying by a ratio) never converges to a finite sum, whatever that constant value (unless it is zero) — only the multiplicative nature of the geometric sequence, combined with a ratio of magnitude less than 1, allows this convergence to a finite sum.
Example: a = 1, r = 0.5 (Zeno's paradox)
Inputs
First term: 1. Ratio: 0.5.
Calculation
S = 1 ÷ (1 − 0.5) = 1 ÷ 0.5 = 2.
Result
The sum of this infinite series (1 + 0.5 + 0.25 + 0.125 + ...) is exactly 2.
Frequently asked questions
Why can the sum of an infinite number of terms be finite?
Because each term becomes smaller and smaller when |r| < 1: the added terms quickly become negligible, and the sum approaches a precise limit without ever exceeding it, a bit like approaching a wall indefinitely without ever quite touching it, but getting closer by an ever more minute distance at each step. It is this convergence behavior, rigorously studied in mathematical analysis, that gives a precise meaning to this apparently paradoxical sum.
What happens if |r| ≥ 1?
The series diverges: if r ≥ 1, each term stays equal to or larger than the previous one, and the sum grows without limit toward infinity. If r ≤ −1, the terms alternate in sign without ever decreasing in absolute value, and the sum oscillates indefinitely without ever settling to a precise value. In both cases, there is no finite sum, which is why this calculator restricts the ratio to the open interval between −1 and 1.
What is the difference from the sum of a finite number of terms?
Our geometric sequence calculator computes the sum of a determined number n of terms, a formula valid for any ratio (including |r| ≥ 1, since the sum always stays finite with a finite number of terms). This calculator, on the other hand, adds infinitely many terms: only in this infinite case does the condition |r| < 1 become necessary to obtain a finite result.