Power series treat functions as polynomials that never end
Add 1, then x, then x squared, then x cubed, forever, and for small enough x you get exactly 1 divided by 1 minus x. Power series like this let mathematicians rebuild sine, the exponential and countless other functions from endless sums of simple powers, each term a little correction to the last.
A power series is an infinite sum in which each term is a coefficient multiplied by a rising whole-number power of the distance from a fixed centre. When the centre is zero, as in a Maclaurin series, the terms are simply powers of x. Cutting the sum off anywhere gives an ordinary polynomial, and conversely any polynomial counts as a power series whose terms eventually all become zero. Negative powers are not allowed, since those produce Laurent series, and fractional powers belong to Puiseux series.
Famous examples include the geometric series, the exponential function, built from powers of x divided by factorials, and sine, whose alternating odd-power terms work for every real number. All are Taylor series, and Borel's theorem implies the reverse too: any power series at all arises as the Taylor expansion of a smooth function.
Convergence is the crucial question. A series always converges at its centre, where only the first term survives, but may diverge everywhere else. Otherwise there is a radius of convergence: inside it the sum settles, outside it blows up. The Cauchy–Hadamard theorem gives a formula for that radius from the coefficients. In the complex plane the region inside is a disc, where convergence is absolute. On the boundary itself nothing general can be said, although Abel's theorem links the value there to limits from inside.
Adding two series around the same centre simply adds matching coefficients, and the result converges at least as widely as the narrower of the two. The idea reaches well beyond analysis: generating functions in combinatorics, the Z-transform in electronic engineering, and p-adic numbers in number theory. Even ordinary decimal notation is a power series with the variable fixed at one tenth.
Source: Power series
