power series
A power series is like a polynomial of infinite length: instead of stopping at some highest power, you keep adding terms a_0, a_1 x, a_2 x^2, a_3 x^3, and so on forever. Each new term is the old variable raised to one higher power, scaled by a number called a coefficient. Many familiar functions — the exponential, sine, cosine, the logarithm — are secretly such infinite polynomials in disguise.
Formally, a power series centered at a point a is a series of the form sum_{n=0}^infinity c_n (x - a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + ... , where the c_n are fixed real (or complex) numbers and x is the variable. The number a is the center, and the c_n are the coefficients. For each fixed x the series is an ordinary infinite series, so it either converges to a number or diverges.
The crucial fact is that the set of x for which the series converges is never arbitrary: it is always an interval (in the real case) or a disk (in the complex case) centered at a, possibly together with some boundary points. Inside that region the series defines a function, and that function is as smooth and well-behaved as one could hope. So a power series is both a series of numbers (for each x) and a recipe for building a function term by term.
sum_{n=0}^infinity x^n / n! converges for every real x and equals e^x. Here a = 0 and c_n = 1/n!, so the coefficients shrink rapidly enough that the series converges everywhere.
The exponential function as a power series centered at 0.
Every polynomial is a power series in which all coefficients past some point are zero. The geometric series 1 + x + x^2 + ... is the simplest non-trivial example; it converges to 1/(1-x) exactly when |x| < 1.