Power Series

Almost any function, locally, is a polynomial of infinite length — and that's how a computer evaluates a sine.

Suggested next → Complex Analysis · MATH · T5

The brief

In the fourteenth century the Indian mathematician Madhava of Sangamagrama — three centuries before Newton, in the Kerala school — discovered that sine, cosine, and arctangent could each be written as infinite sums of powers of their arguments. He used the arctangent expansion to compute π to eleven decimal places. The Kerala results were lost to Europe; when Newton, Gregory, Leibniz, and finally Brook Taylor rediscovered the technique in the seventeenth century, they were rediscovering Madhava's insight: almost any function, near a point, can be approximated arbitrarily well by a polynomial of high enough degree. The polynomials, in the limit, became power series.

The deep claim, codified by Taylor in 1715 and given clean modern form by Cauchy a century later, is that a smooth function near a point is determined by its derivatives there: f(x) = Σ f⁽ⁿ⁾(a)/n! · (x − a)ⁿ encodes the whole local behavior of f in a single sequence of numbers. The exponential, sine, and cosine each converge on the entire real line. Logarithm, geometric series, and most other transcendentals converge inside a finite radius set by the nearest singularity, with the Cauchy-Hadamard formula 1/R = lim sup |cₙ|^{1/n} reading the radius off the coefficients. Smooth is not enough: e^{−1/x²} is smooth at zero with every derivative zero but is not the zero function — the Taylor series there is identically zero while the function is not. Functions for which the Taylor series equals the function are analytic, and the gap between smooth and analytic is one of the foundational distinctions of real analysis.

Power series inherit the algebra of polynomials almost verbatim: within the radius of convergence, addition, multiplication, term-by-term differentiation, and integration all behave as expected. Generating functions exploit this — the Fibonacci generating function 1/(1 − x − x²) and the binomial theorem are the canonical small examples, and the technique scales up to recursions no human could solve in closed form. Laurent series with negative powers classify singularities in the complex plane; analytic continuation extends a function defined locally by a power series to a maximal domain, turning the local derivative data into a global object.

Why nowPower series are everywhere applied calculation happens. When a calculator evaluates sin(0.1) it sums a handful of Taylor terms and rounds; when a finite-difference scheme integrates a differential equation, it is using truncated Taylor expansions of the unknown function. Perturbation theory in physics expands physical quantities in a small parameter — the coupling constant of quantum field theory, 1/c² in post-Newtonian gravity, ℏ in the classical limit of quantum mechanics — and reads off the corrections term by term. Automatic differentiation in modern ML frameworks (PyTorch, JAX) is, in spirit, a machine for computing Taylor coefficients of computational graphs efficiently. The small observation that smooth functions look polynomial up close is among the most-used facts in applied mathematics.

Further readingCalculus (Spivak, 4th ed., 2008). Principles of Mathematical Analysis (Rudin, 3rd ed., 1976). generatingfunctionology (Wilf, 3rd ed., 2006). Analytic Combinatorics (Flajolet & Sedgewick, 2009).