ConverterHub

Infinite Geometric Series Calculator

Find the limit of the infinite series.

Result

24.0000

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.

Try this example →

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.

Was this helpful?

You can pick up to 3 reactions.

Comments

  • Be the first to comment.

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators