Radius and interval of convergence with power series

Get the most by viewing this topic in your current grade. Pick your course now.

Introduction
Lessons
  1. Radius and Interval of Convergence with Power Series Overview
  2. Radius and Interval of Convergence
  3. Checking the Endpoints for the Interval of Convergence
Examples
Lessons
  1. Questions based on Radius of Convergence
    Determine the radius of convergence for the following power series:
    1. n=02nxnn! \sum_{n=0}^{\infty}\frac{2^nx^n}{n!}
    2. n=03nx+32n+1 \sum_{n=0}^{\infty}3^n|x+3|^{2n+1}
  2. Radius of Convergence of Sine and Cosine
    Determine the radius of convergence for the following power series:
    1. n=0(1)nx2n+1(2n+1)! \sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{(2n+1)!}
    2. n=0(1)nx2n(2n)! \sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}
  3. Questions based on Interval of Convergence
    Determine the interval of convergence for the foll owing power series:
    1. n=0n+23n(x+5)n \sum_{n=0}^{\infty}\frac{n+2}{3^n}(x+5)^n
    2. n=0(2x3)nnn \sum_{n=0}^{\infty}\frac{(2x-3)^n}{n^n}
Free to Join!
StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun - with achievements, customizable avatars, and awards to keep you motivated.
  • Easily See Your Progress

    We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.
  • Make Use of Our Learning Aids

    Last Viewed
    Practice Accuracy
    Suggested Tasks

    Get quick access to the topic you're currently learning.

    See how well your practice sessions are going over time.

    Stay on track with our daily recommendations.

  • Earn Achievements as You Learn

    Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.
  • Create and Customize Your Avatar

    Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.
Topic Notes
In this lesson, we will learn about what a power series is. Power series have coefficients, x values, and have to be centred at a certain value a. Our goal in this section is find the radius of convergence of these power series by using the ratio test. We will call the radius of convergence L. Since we are talking about convergence, we want to set L to be less than 1. Then by formatting the inequality to the one below, we will be able to find the radius of convergence. Lastly, we will learn about the interval of convergence. The interval of convergence is the value of all x's for which the power series converge. Also make sure to check the endpoint of the interval because there is a possibility for them to converge as well.
Note *Power Series are in the form:

n=0cn(xa)n \sum_{n=0}^{\infty}c_n(x-a)^n

where cnc_n are the coefficients of each term in the series and aa is number.

To find the Radius of Convergence of a power series, we need to use the ratio test or the root test. Let An=cn(xa)nA_n=c_n(x-a)^n. Then recall that the ratio test is:

L=limL=\limn →\inftyAn+1An|\frac{A_{n+1}}{A_n}|
and the root test is
L=limL=\limn →\inftyAn1n|A_n|^{\frac{1}{n}}

where the convergence happens at LL< 11 for both tests. More accurately we can say that the convergence happens when xa|x-a| < RR, where is the Radius of Convergence.

The Interval of Convergence is the value of all xx's, for which the power series converges. So it is important to also check if the power series converges as well at xa=R|x-a|=R.