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Radius and Interval of Convergence of Power Series

Abigail Young

Abigail Young

7 min read

Next Topic - Finding Taylor or Maclaurin Series for a Function

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Study Guide Overview

This study guide covers power series, including their radius and interval of convergence. It explains how to use the ratio test to determine these values and how to test the endpoints of the interval of convergence. The guide also provides a practice question and a summary of key concepts related to power series convergence.

#10.13 Radius and Interval of Convergence of Power Series

Oooookay, that title definitely had a lot of buzzwords… namely, radius of convergence, interval of convergence, and power series. You haven’t seen them in any of the previous study guides, either, so they’re definitely new to you. Let’s define them one by one! 😉

#👊 What’s a Power Series?

A power series is a series of the form ∑n=0∞an(x−r)n\sum^{\infty}_{n=0}a_n(x-r)^n∑n=0∞​an​(x−r)n, where n is a non-negative integer, ana_nan​ is a sequence of real numbers, and r is a real number.

In this case, ana_nan​ can be any sequence, most of which you’ve already seen in previous study guides! r refers to where we center our power series function (e.g., a “power series centered at x = 3” will give us (x - 3) at that part of the series).

#🔵 Radius and Interval of Convergence

One of the questions we have about power series approximations of functions is where the approximation is valid, or in other words, where the power series converges. For a given x, we can find the radius, and then the interval of convergence for a power series.

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Previous Topic - Lagrange Error BoundNext Topic - Finding Taylor or Maclaurin Series for a Function

Question 1 of 11

Which of the following series is a power series? 🤔

∑n=0∞n22n\sum_{n=0}^{\infty} \frac{n^2}{2^n}∑n=0∞​2nn2​

∑n=0∞(x−2)nn!\sum_{n=0}^{\infty} \frac{(x-2)^n}{n!}∑n=0∞​n!(x−2)n​

∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^p}∑n=1∞​np1​

∑n=1∞(−1)nn\sum_{n=1}^{\infty} \frac{(-1)^n}{n}∑n=1∞​n(−1)n​