Convergence of Complex Series

Complex Analysis · Axiom Academy

LESSON Convergence of Complex Series An infinite sum of complex numbers is a journey of partial sums in the plane — and a power series claims its own disk of convergence. 1. Convergence Is a Path in the Plane A complex series converges when its sequence of partial sums S_N approaches a limit. Take the geometric series with , so . Each new term z^ n is half as long as the last and turned a further , so the partial sums march in ever-shorter, ever-rotating steps — a spiral that curls onto its limit . Replace the constant z by a variable and let the coefficients vary: a power series . Its set of convergence is never ragged — there is a single radius R (the radius of convergence ) such that the series converges for every z inside the disk |z| < R and diverges for every z outside it. Watch a test point travel outward: inside the disk the term sizes |a_n z^n| collapse to 0 ; cross the boundary circle and they blow up. Terms geometrically — the series converges absolutely. Terms do not even tend to 0 , so the series diverges. No universal verdict — it can converge at some points and diverge at others. means it converges everywhere; R = 0 means only at z = 0 . For every coefficient is 1 , the radius is R = 1 , and inside that unit disk the sum is exactly — the building block of every power series. 3. The Ratio Test Measures the Radius

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