---
title: "Power Series and the Radius of Convergence"
description: "Read a power series as a function defined where it converges, find its radius with the ratio test, and check the endpoints separately."
canonical: https://lightmysky.com/learn/mathematics/power-series-and-the-radius-of-convergence-mt_RnYb0JLKbD
source: https://lightmysky.com/learn/mathematics/power-series-and-the-radius-of-convergence-mt_RnYb0JLKbD.md
retrieved: 2026-09-12
---

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# Power Series and the Radius of Convergence

Read a power series as a function defined where it converges, find its radius with the ratio test, and check the endpoints separately.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/power-series-and-the-radius-of-convergence-mt_RnYb0JLKbD

## Ready when they can

- Find the radius of convergence of a given power series
- Test both endpoints and state the full interval
- Differentiate or integrate a power series term by term inside its radius

## Lesson: How far a power series reaches

A power series is an infinite polynomial built around a center, like c0 + c1 x + c2 x squared and on. It converges for x close to the center and diverges for x far away. The distance from the center to that boundary is the radius R. In extreme cases R can be 0, when only the center works, or infinite, when factorial denominators beat every growth.

**Example.** Take the sum of x to the n over n. The ratio of consecutive absolute terms simplifies to the absolute value of x, which stays below 1 exactly when x sits within 1 of 0, so R is 1. At x = 1 you get the harmonic series, which diverges, and at x = minus 1 you get the alternating harmonic series, which converges. So the full interval runs from minus 1 inclusive to 1 exclusive.

Inside its radius you may differentiate or integrate a power series term by term, and the radius stays exactly the same. Only the endpoints may change fate, so you test them again after the operation. For example, differentiating or integrating a series with radius 1 gives a new series that still has radius 1.

**Tip.** Work radius first and endpoints second. Find R with the ratio test, then test each endpoint on its own with your series tools, then write the bulk plus whichever endpoints pass.

**Recap.** Radius first with the ratio test, endpoints second with series tools, then the bulk plus passing endpoints.

## Practice

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

## Needs first

- [Infinite Series and the Geometric Series](https://lightmysky.com/learn/mathematics/infinite-series-and-the-geometric-series-mt_EPE-ERFFQt)
- [Alternating Series and Absolute Convergence](https://lightmysky.com/learn/mathematics/alternating-series-and-absolute-convergence-mt_sk6F4n1USH)

## Opens up

- [Pointwise and Uniform Convergence](https://lightmysky.com/learn/mathematics/pointwise-and-uniform-convergence-mt_iAGk1ML5MA)
- [Power Series and Analyticity in the Complex Plane](https://lightmysky.com/learn/mathematics/power-series-and-analyticity-in-the-complex-plane-mt_Mwxapy_pU8)
- [Taylor and Maclaurin Series](https://lightmysky.com/learn/mathematics/taylor-and-maclaurin-series-mt_SgI9Pn9RIO)
- [Generating Functions](https://lightmysky.com/learn/mathematics/generating-functions-mt_WAX3O-HjRk)