An infinite geometric series keeps adding terms forever: a + ar + ar2 + ar3 + ···. Sometimes the running total settles on a single number; sometimes it races off to infinity. The whole outcome hinges on one value — the ratio r. Drag it across 1 and watch.
Running total after each term
Each dot is a partial sum — the total after that many terms. The dashed line is the limit the totals chase.
Infinite geometric seriesdrag the sliders to begin
Inside the green band the sum converges. Cross either edge — at r = −1 or r = 1 — and it diverges.