a complex power series
A power series is an infinite polynomial: instead of stopping at some highest power, you keep going forever. Centred at a point z_0 it is sum a_n (z - z_0)^n = a_0 + a_1 (z - z_0) + a_2 (z - z_0)^2 + ..., where the coefficients a_n are fixed complex numbers and z is a variable point in the plane. For each chosen z this is an ordinary complex series whose terms you can add up — provided it converges. So a power series is really a recipe: feed it a z, and if the series converges, out comes a single complex number, the value of the function it defines.
Where does it converge? Remarkably, the answer always has the same shape: there is a number R between 0 and infinity, the radius of convergence, such that the series converges (absolutely) for every z with |z - z_0| < R and diverges for every z with |z - z_0| > R. The set where it converges is a disk centred at z_0. This is not obvious — there is no reason in advance the convergence region should be round — but it falls straight out of comparison with a geometric series. The radius R can be 0 (the series converges only at the centre and is useless), a finite positive number, or infinity (the series converges in the whole plane).
Power series matter because they are the local genetic code of complex analysis. The central theorem of this part of the subject says that every holomorphic function, near any point of its domain, IS a convergent power series — its Taylor series. So power series are not a special family of functions to study separately; they are what every nice complex function looks like up close. And because a power series can be differentiated and integrated term by term inside its disk, manipulating holomorphic functions reduces, locally, to manipulating these infinite polynomials.
The geometric series sum z^n = 1 + z + z^2 + ... has all a_n = 1 and centre z_0 = 0. It converges exactly when |z| < 1, where its sum is the simple closed form 1 / (1 - z). Outside the unit disk it diverges, even though 1 / (1 - z) is a perfectly good function there — the series only captures the function inside its disk.
The series sum z^n and the function 1/(1 - z) agree only on |z| < 1; the radius 1 is the distance from the centre 0 to the singularity at z = 1.
A power series defines its function only inside its disk; the same function may be defined far beyond, but you reach those regions by re-expanding around a new centre (analytic continuation), not by the original series.