Walk halfway to the wall, then half the remaining distance, then half again — logic insists you never arrive, yet your hand touches the wall. In plain words, an infinite geometric series converges when each term shrinks by a fixed ratio smaller than 1, and the total equals the first term divided by one minus the ratio: S = a / (1 − r). Adding 1 + 1/2 + 1/4 + 1/8 + ... forever never passes 2, because 1 / (1 − 1/2) = 2. Zeno's ancient paradox of the runner halving the distance dissolves into the same arithmetic. Finance students pricing a perpetuity, programmers summing a converging loop, and anyone proving that 0.999... equals 1 use this sum. Infinity is only scary until the ratio drops below one.

This calculator sums the geometric case. Type the first term into the Variable A box, say 1, and the common ratio into the Variable B box, say 0.5. Read the infinite sum from the Result box: 2. Keep the ratio between −1 and 1 — outside that range the series diverges and no finite sum exists.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Infinite Series Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.