10.2 Working with Geometric Series

This whole unit is BC only. A geometric series has a constant ratio between consecutive terms: a + ar + ar² + ar³ + … = Σ (n = 0 to ∞) arⁿ. The number a is the first term and r is the common ratio. To find r, divide any term by the one before it.

The partial sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) when r ≠ 1. If |r| < 1, then rⁿ → 0, so

Σ (n = 0 to ∞) arⁿ = a/(1 − r), for |r| < 1.

If |r| ≥ 1 (and a ≠ 0), the terms don't approach 0, so the series diverges.

A reliable way to remember the formula: sum = (first term)/(1 − ratio). “First term” means whatever the first term actually is, which matters when the index doesn't start at 0.

• Combine powers: 2ⁿ⁺¹/5ⁿ = 2 · (2/5)ⁿ, so r = 2/5.
• Watch negative ratios: Σ (−1/3)ⁿ has r = −1/3, and |r| < 1, so it converges.
• Find the first term by plugging in the starting index, not by assuming n = 0.
• Repeating decimals are geometric: 0.272727… = 27/100 + 27/100² + … = (27/100)/(1 − 1/100) = 27/99 = 3/11.

A ball dropped from 10 feet rebounds to 3/4 of its previous height each time. The total distance it travels is 10 feet down, then up and down for each bounce: 10 + 2(7.5 + 5.625 + …). The bounces form a geometric series with first term 7.5 and ratio 3/4, so the total is 10 + 2 · 7.5/(1 − 3/4) = 10 + 60 = 70 feet. Notice the first drop is counted once and every rebound twice.

Geometric series are the main kind of series whose exact sum you can find with a simple formula. (Telescoping series, like the one in 10.1, are another.) They're also the main benchmark for comparison tests (10.6), the reason the ratio test works (10.8), and the starting point for power series: 1/(1 − x) = 1 + x + x² + x³ + … for |x| < 1 (10.14, 10.15).