Radius and interval of convergence with power series

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  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
  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}
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.