Power Series Applications

Real Analysis · Axiom Academy

Some integrals have no formula and some functions have no shortcut — so we trade them for an infinite sum and take the first few terms. Watch power series do real work. A power series turns a hard function into infinitely many easy ones — and you only ever add up a handful. Here are three places that single move cracks a problem nothing else can. Compute an integral with no formula The bell-curve area ∫₀¹ e −x² dx has no antiderivative in elementary functions — no substitution will ever finish it. So expand e −x² as a power series and integrate it one term at a time. Add terms and watch the running total close in on the true value. A chip can only add and multiply — it can't “know” e x . It runs the Taylor series e x = Σ x k /k!. Because each denominator is a factorial , the terms collapse fast: a handful nails e = 2.71828… to machine precision. Add terms and watch the digits lock in. The shortcut physics runs on: sin θ ≈ θ The series sin θ = θ − θ³/6 + θ⁵/120 − … means that for small θ the very first term, just θ, is almost the whole answer — the famous small-angle rule behind pendulums and optics. Drag the angle and watch where “sin θ ≈ θ” is safe, and where it breaks. One move, three payoffs: integrate the unintegrable (∫ e −x² ), evaluate to machine precision (every e x , sin, cos your calculator returns), and simplify the messy (sin θ ≈ θ). Swap a hard function for its power series, keep a few terms, and intractable problems become arithmetic.

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