The Infinite Polynomial
A Power Series is a polynomial with an infinite number of terms. This is a revolutionary idea: it means that complex functions like \(\sin(x)\) or \(e^x\) can be written as a simple list of \(x^n\) terms.
\[f(x) = \sum c_n x^n = c_0 + c_1x + c_2x^2 + \dots\]
Convergence
A power series only "works" within its Radius of Convergence. Inside this radius, the infinite sum equals the function. Outside, the sum diverges and is meaningless.
Worked Examples
Example 1: The Geometric Power Series
The function \(f(x) = \frac{1}{1 - x}\) can be written as the power series \(1 + x + x^2 + x^3 + \dots\).
- If \(x=0.1\), then \(1/(1-0.1) = 1.111\dots\).
- The series \(1 + 0.1 + 0.01 + 0.001 = 1.111\).
- They are identical!
The Bridge to Quantum Mechanics
Computers do not "know" what a sine wave is. They only know how to add and multiply. When a physics simulation calculates a quantum wave, it is using the first few terms of a power series. In Quantum Mechanics, we often find ourselves with a differential equation that we cannot solve. Our "Plan B" is always to assume the answer is a power series and solve for the coefficients \(c_n\). This is how we solved the Quantum Harmonic Oscillator (Chapter 13)—the single most important model in physics.