AP® Calculus BC · Unit 10 · Topic 10.2
Working with Geometric Series
A geometric series adds terms that each equal the last one times a fixed ratio r. If |r| < 1, the infinite sum settles on the exact value a/(1-r), where a is the first term.
Picture walking toward a wall, each step covering half the remaining distance. You never touch the wall, but the total distance walked gets closer and closer to a fixed number. That is what a convergent geometric series does: it adds infinitely many terms, each a constant ratio r times the one before, and the partial sums home in on one finite value as long as |r| < 1. If |r| ≥ 1 the terms stop shrinking fast enough, and the sum either grows without bound or oscillates forever.
The formula comes from a clean trick. Write S as the sum a + ar + ar2 + ar3 + ... Multiply every term by r to get rS = ar + ar2 + ar3 + ... Subtract: S - rS lines up so that every term except the very first a cancels out. That leaves S(1-r) = a, so S = a/(1-r). This works for a finite sum too, but letting the series run forever is only safe when |r| < 1, since otherwise the terms you threw away do not shrink to zero.
This is the gateway to the rest of the unit. Every convergence test you meet later is really asking whether a series behaves enough like a geometric one to settle down. Power series and Taylor series, which rebuild functions like 1/(1-x) or ex, are built by stacking geometric-style terms with a variable ratio, so mastering a/(1-r) now makes those later topics read as familiar patterns instead of new tricks.
Key terms
- geometric series
- A sum of terms where each term is the previous one multiplied by a fixed ratio r.
- common ratio (r)
- The fixed number you multiply by to move from one term to the next.
- convergence
- The property that partial sums approach a single finite value as more terms are added.
- partial sum
- The sum of only the first several terms of a series, before taking the infinite limit.
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End of topic.