Geometric Series

Core Idea

A geometric series has the form sum from n=0 to infinity of a*rⁿ = a/(1 - r), converging if and only if |r| < 1. It is the most important series because it has a known closed-form sum, serves as a benchmark for comparison tests, and is the basis for power series and Taylor series. The partial sum formula S_N = a(1 - r^N)/(1 - r) shows exactly how the series converges.

How It's Best Learned

Derive the partial sum formula by multiplying S_N by r and subtracting. Take the limit as N -> infinity to get the infinite sum. Practice identifying geometric series in various forms (e.g., sum of (2/3)ⁿ, sum of (-1)ⁿ / 4ⁿ). Apply to repeating decimals and real-world scenarios.

Common Misconceptions

• Applying the formula when |r| >= 1 (the series diverges in this case).
• Off-by-one errors on the starting index (the formula changes if the series starts at n = 1 vs. n = 0).
• Not recognizing a geometric series when it is written in a non-standard form.

Explainer

A geometric series is one where each term is obtained by multiplying the previous one by a fixed ratio r. You already know geometric sequences from earlier work; a geometric series is simply the sum of such a sequence. The central question is: when you add infinitely many terms, can the total be finite?

The answer depends entirely on |r|. To see why, consider the partial sum S_N = a + ar + ar² + ... + ar^N. Multiply both sides by r: rS_N = ar + ar² + ... + ar^N+1. Subtract the second equation from the first and almost everything cancels, leaving S_N(1 - r) = a(1 - r^N), so S_N = a(1 - r^N)/(1 - r). As N → ∞, the term r^N → 0 only when |r| < 1 — when r is a fraction between -1 and 1. In that case the infinite sum collapses to the clean formula a/(1 - r). If |r| ≥ 1, the terms don't shrink and the sum grows without bound.

The starting index matters more than students expect. The formula a/(1 - r) uses the first term actually in the sum as a. If your series starts at n = 0, the first term is a·r⁰ = a. If it starts at n = 1, the first term is a·r¹. Plugging in the wrong first term is the most frequent source of errors, so always identify a by evaluating the term at the lowest index before applying the formula.

Geometric series are foundational to the rest of Calculus 2 because power series — and ultimately Taylor series — are geometric series with r replaced by a variable expression. The radius of convergence of a power series is precisely the set of x-values for which the underlying geometric series converges. Every Taylor series you will encounter is, in a structural sense, built on the geometric series formula. Getting comfortable recognizing and summing geometric series now will pay dividends through the end of the course.