How do I find the sum of an infinite geometric series? When does it converge?
I know the formula S = a/(1-r) for infinite GP. But my textbook says this only works when |r| < 1. Why does the series not converge when |r| >= 1? Can you give a physical example?
1 Answer
For an infinite GP with first term a and ratio r: S = a/(1-r), valid ONLY when |r| < 1 (r between -1 and 1). When |r| < 1: each term gets smaller, so the total sum approaches a finite limit. Example: 1 + 1/2 + 1/4 + 1/8 + ... = 1/(1-0.5) = 2. When |r| >= 1: terms do not shrink, sum grows without bound (diverges). Example: 1 + 2 + 4 + 8 + ... never reaches a finite sum. Physical example: A ball dropped from 1m bounces to 0.6m each time. Total distance = 1 + 2*(0.6 + 0.36 + ...) = 1 + 2*(0.6/(1-0.6)) = 1 + 3 = 4m. The series converges because the bounce ratio 0.6 < 1.
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