Infinite Geometric Series Calculator

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.

This calculator takes the first term and common ratio of a convergent series, where the ratio's absolute value is less than 1, and returns the exact value the infinite sum converges to.