Interactive visualization of geometric series convergence

Add infinitely many numbers and get a finite answer

When a geometric series has a ratio between −1 and 1, adding infinite terms converges to a single value. The smaller the ratio, the faster the series settles. Drag the slider to change the ratio and watch the partial sums converge toward the limiting sum.

Convergence visualiser
r = 0.5
Drag to change the ratio r — watch the series approach its limit
Partial sum S₁₀ = 1.999
Common ratio r
−0.95 0 0.95
Limiting sum S∞
2.000
Terms needed (99%)
7
The limiting sum formula comes from the partial sum formula Sn = a(1 − rn)/(1 − r). When |r| < 1, the term rn shrinks toward zero as n increases, because multiplying a fraction by itself repeatedly makes it vanishingly small. So as n → ∞, the expression 1 − rn approaches 1 − 0 = 1, leaving S = a/(1 − r). This is why the ratio must sit strictly between −1 and 1: outside that interval, rn grows instead of shrinking, the partial sums spiral away, and no finite limit exists.
Know This
A geometric series converges to S = a/(1−r) if and only if |r| < 1, because only then does rn → 0 as the number of terms approaches infinity.