Geometric Series Convergence Checker

About this calculator

An infinite geometric series a + ar + ar² + ar³ + … converges to a finite sum only when the common ratio satisfies |r| < 1; in that case the sum equals a ÷ (1 − r). If |r| ≥ 1, the terms don't shrink fast enough (or grow), and the series diverges to infinity or oscillates without settling on a value.

This checker takes the first term and common ratio, tests the |r| < 1 condition, and returns the exact sum when convergence holds. Calculus and pre-calculus students use it to check series-convergence homework, while it also mirrors real applications such as computing the present value of a perpetuity in finance, the total distance in a bouncing-ball or Zeno's-paradox problem, or the steady-state gain of a feedback loop in engineering — all of which reduce to the same a ÷ (1 − r) formula.