Key Idea: Some never-ending sums still add up to a finite total. A geometric series has a sum to infinity only when −1 < r < 1. In real models it is the long-run total or steady level.
Sum to infinity in four steps
- Gate — If |r| ≥ 1, write "no sum to infinity".
- Formula — S∞ = u₁/(1 − r).
- Backwards — Given S∞ and u₁, solve S∞(1 − r) = u₁ for r.
- Context — Say what the total means, for example a steady drug level.
| Part | Distance |
|---|---|
| First drop | 3 m |
| Rebounds: u₁ = 1.8, r = 0.6 | 1.8/0.4 = 4.5 m |
| Each rebound goes up and down | 2 × 4.5 = 9 m |
| Total | 12 m |
Check r first. Only a ratio between −1 and 1 lets the sum settle.
Does 1 + 2 + 4 + 8 + … have a sum to infinity? No. Here r = 2, which fails the gate.
Why is the 12 m total a slight over-estimate? The model lets the ball bounce forever. A real ball stops after a few bounces.
S∞ = 10 and u₁ = 4. Find r. 10(1 − r) = 4, so r = 0.6.
Why count each rebound height twice? The ball goes up that height and then comes down it.