Series in the Complex Plane

Complex Analysis · Axiom Academy

Writing a function as an infinite sum — and watching where that sum lives A function, rebuilt as an endless sum One of the most powerful ideas in analysis is to stop treating a function as a formula and start treating it as an infinite sum of simple powers. In the complex plane this is even sharper than on the real line: the central question is when we can write Watch the geometric series assemble itself. Each new term adds a partial sum , and the shaded region of the plane where those partial sums lock onto grows — and settles into a perfect disk |z| < 1 . The partial sums converge exactly inside the disk and blow up outside it — the series is the function, but only where it lives. What stops the disk: the nearest singularity Here is the beautiful part, and it is unique to complex analysis. The radius of convergence is not arbitrary — it reaches out from the center exactly to the nearest place the function breaks. Drag the singularity around the plane and watch the disk's radius track its distance: . Push the singularity straight up to i and the radius is still 1 — a break the real line never sees can still cap a real series. That is why stops at |x| = 1 even though looks perfectly smooth. Two kinds of series, two kinds of region

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