Taylor Series Convergence

Differential Equations · Axiom Academy

LESSON Taylor Series Convergence Every power series solution has a radius of convergence — and for a differential equation, that radius is set by the nearest singular point of its coefficients. 1. Radius of Convergence — a Quick Review A power series centered at x = x_0 has the form shown below. It converges for every x inside a radius R of x_0 — the interval (or, in , disk) of convergence — and diverges outside it. On the boundary |x-x_0|=R itself, anything can happen; that edge case needs a separate test. 2. Singularities Set the Limit Now put a differential equation in standard form, shown below. The functions P(x) and Q(x) are its coefficients . If they have singularities — points where they blow up or are undefined — those points cap how far our series solution can reach. Why the complex plane? A function can look perfectly smooth along the real line and still have a hidden singularity just off it. Take : real-valued and totally smooth for every real x , yet its Maclaurin series only converges for . The reason: 1+x^2=0 at , each a distance 1 from the origin — invisible on the real axis, but the series can't outrun it. 3. Airy's Equation — When R Is Infinite Airy's equation is a workhorse of optics and quantum mechanics. Written in standard form, its coefficients are P(x)=0 and Q(x)=-x — both plain polynomials. Built from the recurrence the coefficients generate, this series converges for all real (and complex) x — exactly as the theorem guarantees.

This is the written version of the interactive lesson above. See the full Differential Equations course.