A geometric series multiplies by a constant ratio r each step.
Try this
4
Sₙ = 1 + ½ + ¼ + ⅛ + … (a = 1, r = ½) → 2 = 1.88
When does it converge?
- If |r| < 1, the terms shrink and Sₙ approaches a/(1 − r) — a finite limit.
- If |r| ≥ 1, the terms don't shrink, so the sum runs off to infinity (or oscillates).
- Drag the slider above: each step adds half the remaining gap to 2, but never quite arrives.
Your turn
Find 3 + 6 + 12 + 24 + 48.
0.999… = 9/10 + 9/100 + 9/1000 + … is a geometric series with a = 9/10, r = 1/10. Its sum is (9/10)/(1 − 1/10) = 1 exactly.
Sum of n terms
Infinite sum (|r|<1)
Try it
1 + 1/2 + 1/4 + 1/8 + ...
a=1, r=1/2. S = 1/(1−1/2) = 2.