interval of convergence

If the radius of convergence tells you how wide the zone of good behavior is, the interval of convergence is the actual list of real inputs you are allowed to plug in. It is the open zone given by the radius, plus a final decision about each of the two endpoints — included or not.

For a real power series sum c_n (x - a)^n with radius R (assume 0 < R < infinity), the interval of convergence is one of the four intervals (a - R, a + R), [a - R, a + R), (a - R, a + R], or [a - R, a + R], depending on whether the series converges at the left endpoint a - R, the right endpoint a + R, both, or neither. When R = infinity the interval is the whole real line; when R = 0 it is just the single point {a}.

Determining the interval is a two-step job. First find R, usually by the ratio or root test, which settles everything in the open interval. Then test each endpoint separately by substituting it in and applying ordinary series tests — the alternating series test, comparison, p-series, and so on. The endpoints are genuinely independent: a series can converge at one and diverge at the other.