the radius of convergence
/ RAY-dee-us /
Every power series sum a_n (z - z_0)^n has a built-in reach: a number R, from 0 to infinity, that says how far out from the centre z_0 the series still makes sense. Inside the circle of radius R the series converges and gives you a function; outside that circle it diverges and gives you nothing. R is called the radius of convergence, and it is determined entirely by the coefficients a_n — it does not depend on which z you test.
Concretely, R is the largest radius such that the series converges for all z with |z - z_0| < R. You can compute it from the coefficients in two equivalent ways. The root-test formula (Cauchy-Hadamard) says 1/R = limsup |a_n|^(1/n). The ratio-test shortcut, valid when the limit exists, says 1/R = lim |a_(n+1)| / |a_n| — so R = lim |a_n| / |a_(n+1)|. Both read off how fast the coefficients grow or shrink: coefficients that decay fast give a large R; coefficients that grow give a small R. Three regimes occur: R = 0 means the series converges only at z_0 (a degenerate, useless series); R = infinity means it converges in the entire plane (the function is entire, like e^z); and a finite positive R means a genuine disk of validity.
The radius carries deep geometric meaning, not just a convergence bookkeeping. For a Taylor series of a holomorphic function, R equals the distance from the centre z_0 to the nearest point where the function misbehaves — its nearest singularity. This is why expanding 1/(1 + z^2) around 0 gives radius exactly 1, even though the function looks perfectly smooth on the whole real line: the singularities sit off the real axis at z = i and z = -i, distance 1 away, and the complex series feels them. The radius is the series quietly reporting where the trouble is.
For sum z^n / 2^n the ratio |a_(n+1)| / |a_n| = (1/2^(n+1)) / (1/2^n) = 1/2, so R = 2. The series converges inside the disk |z| < 2 and diverges outside it. (Its sum there is 1 / (1 - z/2), whose only singularity is at z = 2, exactly distance 2 from the centre.)
Coefficients decaying like 1/2^n give radius 2, and the nearest singularity sits exactly at distance 2 — the two always match for a Taylor series.
R tells you nothing about behaviour ON the boundary circle |z - z_0| = R: there the series may converge at every point, no point, or some points. The clean dichotomy (converge inside, diverge outside) deliberately leaves the rim undecided.