power series

A power series is like an endless polynomial: instead of stopping at x^2 or x^3, you keep going with higher and higher powers of x forever, each with its own coefficient. You can picture it as a recipe that builds a function out of infinitely many simple polynomial pieces, layered one atop the next.

It has the form sum c_n*x^n = c_0 + c_1*x + c_2*x^2 + c_3*x^3 + ..., where the c_n are fixed numbers. For a given x this is just an infinite series of numbers, so we ask whether it converges. Typically it converges for all x within some distance R of the center and diverges beyond it; R is called the radius of convergence. So a power series defines a function on an interval, often (-R, R), with the endpoints needing separate checking.

Power series matter because, on their interval of convergence, they behave almost like polynomials — you can differentiate and integrate them term by term — yet they can represent far richer functions. This is what lets calculators and computers compute things like e^x or sin(x): they add up the first several terms of a power series to whatever accuracy is needed.