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Infinite Geometric Series Calculator

Find the limit of the infinite series.

How it is calculated

The tool gives the sum of a geometric series that goes on forever: S = a / (1 − r), where a is the first term and r the common ratio. This only makes sense when the terms shrink toward zero, that is when |r| < 1. If the ratio is 1 or more in size (including −1 and 2), the series does not settle on a number, and the tool returns no result. The first term can be any real number. A negative ratio such as −0.5 is accepted: the terms alternate in sign and the sum is smaller than a. A ratio of 0 simply returns a. The result is shown with four decimals.

Worked example

Start a series at 12 with ratio 0.5: 12 + 6 + 3 + 1.5 + … First compute 1 − r = 1 − 0.5 = 0.5. Then divide the first term: 12 / 0.5 = 24. The calculator shows 24.0000. The partial sums confirm it: 12, 18, 21, 22.5, 23.25 … creep up toward 24 but never pass it.

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Common mistakes

  • Entering a ratio of 1 or more, or −1 or less, expecting a total. The series diverges, so the tool gives no result.
  • Using a percentage as the ratio. A 50 % shrink per step is r = 0.5, not 50.
  • Entering the common ratio as the amount lost. If a bounce loses 30 % of its height, the ratio is 0.7.
  • Forgetting that a negative ratio flips the sign of every second term, which lowers the sum rather than raising it.
  • Treating the result as the sum of a finite number of terms. It is the limit that the partial sums approach but never reach.

About this calculator

An infinite geometric series converges to a finite sum only when the absolute value of its common ratio r is less than 1, in which case the sum equals a divided by (1 − r), where a is the first term — a result derived by taking the limit of the finite geometric series formula as the number of terms approaches infinity, since r raised to increasingly large powers shrinks toward zero.

This convergence formula explains phenomena that seem paradoxical at first glance, like Zeno's dichotomy paradox, where an infinite sequence of ever-smaller distances sums to a finite total, and has real applications in calculating the total distance traveled by a bouncing ball that loses a fixed fraction of height each bounce, the present value of a perpetuity in finance, and converting repeating decimals into exact fractions.

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