NapkinCalc

Calculus 2B — Series & Taylor

Geometric series — infinite sums with finite answers

ELI5: add 1 + ½ + ¼ + ⅛ + ⋯ forever and you do not get infinity — each term covers half the remaining gap to 2. Any geometric series (each term a fixed ratio r times the last, with |r| < 1) converges to 1/(1 − r).

S20:=Sum("0.5i","i",0,20)S_{20} := Sum("0.5^i", "i", 0, 20) = 2.00002.0000 21 terms in — already 1.9999995
✓ pass abs(S20−2)<0.00001\mathrm{abs}\left(S_{20} - 2\right) < 0.00001 converging to exactly 2 = 1/(1 − ½)

Real-world hook: geometric series price loans and annuities, total a bouncing ball's distance, size the medicine that builds up over repeated doses, and explain why 0.999… = 1.

Try it yourself: what does 1 + ⅓ + ⅑ + ⋯ (ratio r = 1/3) sum to? (Use 1/(1 − r).)

geoyou=11−13geo_{you} = \frac{1}{1 - \frac{1}{3}} = 1.50001.5000 ✏️ Your turn: sum the series with ratio 1/3. The check verifies sum × (1 − r) = 1 — so it never states the total.
✓ pass abs(geoyou⋅(1−13)−1)<0.000001\mathrm{abs}\left(geo_{you} \cdot \left(1 - \frac{1}{3}\right) - 1\right) < 0.000001 green when your sum satisfies sum × (1 − r) = 1