What a Power Series Is and Where It Lives

An infinite polynomial centered at a

A power series is an expression sum from n=0 to infinity of c_n (x - a)^n, where the numbers c_n are fixed coefficients and a is the center. For each fixed x it is just an infinite series of real numbers, so it means exactly what every series means: form the partial sums s_N(x) = sum from n=0 to N of c_n (x - a)^n, and ask whether s_N(x) approaches a limit as N grows. If it does, the series converges at x and its sum is a number; if not, it diverges there.

So a power series is not one number; it defines a function on exactly the set of x where it converges. The whole subject begins by pinning down that set. Notice it always contains the center: at x = a every term with n >= 1 vanishes, leaving just c_0.

The convergence set is an interval

Here is the structural fact that makes the theory clean. If a power series centered at a converges at some point x_1 with |x_1 - a| = r_1 > 0, then it converges absolutely at every x with |x - a| < r_1. The reason is a comparison with a geometric series: convergence at x_1 forces the terms c_n (x_1 - a)^n to be bounded, and the closer points get a geometric factor strictly below 1.

Claim: if sum c_n (x_1 - a)^n converges, then for any x with
       |x - a| < |x_1 - a| the series sum c_n (x - a)^n converges absolutely.

Step 1 (terms are bounded).  A convergent series has terms -> 0,
        so the sequence c_n (x_1 - a)^n is bounded:
        there is M with |c_n (x_1 - a)^n| <= M  for all n.

Step 2 (set the ratio).  Let t = |x - a| / |x_1 - a|.
        By assumption 0 <= t < 1.

Step 3 (geometric bound).  For each n,
        |c_n (x - a)^n| = |c_n (x_1 - a)^n| * t^n <= M * t^n.

Step 4 (compare).  sum M t^n is a geometric series with ratio t < 1,
        hence converges.  By the comparison test,
        sum |c_n (x - a)^n| converges -> absolute convergence.   QED

This lemma forces the convergence set to be an interval centered at a. Define R to be the supremum of all distances |x - a| at which the series converges (allowing R = 0 or R = infinity). Then it converges for every |x - a| < R and diverges for every |x - a| > R. That half-width R is the radius of convergence — the single most important number attached to a power series.

The three sizes of R

1. R = 0: the series converges only at the center. Example: sum n! x^n. For any x != 0 the terms n! x^n blow up, so it converges nowhere except x = 0.
2. 0 < R < infinity: converges on (a - R, a + R), diverges outside. Example: sum x^n has R = 1.
3. R = infinity: converges for every real x. Example: sum x^n / n! (the exponential series) converges everywhere.