How to Use the Infinite Geometric Series Calculator
A geometric sequence multiplies each term by a fixed ratio to get the next one. When that common ratio's absolute value is less than 1, adding up infinitely many terms still settles on a finite number — the sum of the infinite geometric series. The formula is S = a₁ / (1 - r), where a₁ is the first term and r is the common ratio. Enter both values and this calculator checks convergence and computes the sum instantly.
For example, a series starting at 4 with a ratio of 0.5 runs 4, 2, 1, 0.5, 0.25, and keeps shrinking, converging to 8. Indeed, S = 4 / (1 - 0.5) = 8 confirms it. On the other hand, if the ratio's absolute value is 1 or greater, the terms grow or oscillate forever and the sum diverges to infinity — in that case the calculator shows a clear divergence message instead of a misleading number.
Infinite geometric series show up in surprisingly practical places: converting repeating decimals into fractions, computing the total area of a fractal, or finding the total distance traveled by a bouncing ball that loses a fixed fraction of height on each bounce. Even with a negative ratio — which produces an alternating series — the same formula gives the exact converged value as long as the absolute value stays under 1.
Frequently Asked Questions
Yes. A negative ratio creates an alternating series, and it still converges as long as its absolute value is less than 1.
The series diverges and has no finite sum, so the calculator shows a divergence message instead of a result.
Every term is 0, so the sum is also 0.