---
title: "Power Series and Analyticity in the Complex Plane"
description: "Expand an analytic function as a power series on a disc, and read the radius of convergence off the nearest singularity."
canonical: https://lightmysky.com/learn/mathematics/power-series-and-analyticity-in-the-complex-plane-mt_Mwxapy_pU8
source: https://lightmysky.com/learn/mathematics/power-series-and-analyticity-in-the-complex-plane-mt_Mwxapy_pU8.md
retrieved: 2026-09-12
---

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# Power Series and Analyticity in the Complex Plane

Expand an analytic function as a power series on a disc, and read the radius of convergence off the nearest singularity.

Subject: Mathematics · Area: Complex Analysis · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/power-series-and-analyticity-in-the-complex-plane-mt_Mwxapy_pU8

## Ready when they can

- Expand a function about a point and state the disc on which the series is valid
- Explain a real series' radius of convergence by pointing at a complex singularity
- Show that a zero of an analytic function is isolated unless the function is identically zero

## Lesson: How far a power series reaches

A power series about a point converges on a disc centered there. The radius marks the distance to the nearest singularity, even one off the real line. Inside the disc you may add, multiply, differentiate, and integrate term by term, and the derived series keeps the same radius.

**Example.** The geometric series 1 plus z plus z squared sums to 1 over (1 minus z) and converges exactly inside the unit disc, since the singularity at 1 sits one unit out. The real function 1 over (1 plus x squared) looks calm at x equal 1, yet its Taylor series about 0 stops at radius 1 because of poles at i and minus i.

Partial sums show the convergence in action. At 0.5, the sum 1 plus 0.5 plus 0.25 plus 0.125 is 1.875, already near the true 2. The discarded tail from the fourth power on is geometric with first term 0.0625 and ratio 0.5, summing to 0.125.

**Tip.** Zeros of a nonzero analytic function stand apart: accumulating interior zeros would force the whole function to zero. Integer multiples of pi escape to infinity, so sine gets away with them. To read a radius off coefficients, use the ratio or root test and take the reciprocal: a limit of 2 means radius 0.5.

**Recap.** The nearest complex obstacle sets the disc, and inside it series behave like well trained polynomials.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Cauchy's Integral Formula and Derivatives of Every Order](https://lightmysky.com/learn/mathematics/cauchys-integral-formula-and-derivatives-of-every-order-mt_mViFwrFNWQ)
- [Power Series and the Radius of Convergence](https://lightmysky.com/learn/mathematics/power-series-and-the-radius-of-convergence-mt_RnYb0JLKbD)
- [Taylor and Maclaurin Series](https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO)

## Opens up

- [Laurent Series and Classifying Isolated Singularities](https://lightmysky.com/learn/mathematics/laurent-series-and-classifying-isolated-singularities-mt_WoQV0gD1O-)