Geometric Series as Power Series

Calculus 2 · Axiom Academy

LESSON Geometric Series as Power Series One series, , becomes a whole library — through substitution, differentiation, and integration. 1. The Foundation: One Series to Build From Add up powers of x forever. Inside the band the terms shrink toward zero and the sum collapses to a single clean fraction — that exact match is what makes this series so useful. Valid only for (radius of convergence R=1 ) 2. Substitution: Swap In a New Expression Anywhere the template has an x , drop in something else. The algebra is free — wherever the new expression stays inside the unit interval, the series still converges. The series we'll integrate into . 3. Differentiation: A Steeper New Series A power series can be differentiated one term at a time. Each x^n becomes , and the radius of convergence is unchanged — so we get a brand-new series for free. Integrating term by term is the most powerful move — it reaches functions powers alone can't. Since , integrating the series for hands us the logarithm. 5. The Arctangent Series and a Formula for Now chain two moves. Substitute -x^2 into the template, then integrate — because , the result is one of the most beautiful series in calculus. You've turned one geometric series into a whole family — and watched each new series lock onto its function inside the interval of convergence. Scroll up to revisit any step.

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