5.16 Radius and interval of convergence with power series
Radius and interval of convergence with power series
Lessons
Notes:
Note *Power Series are in the form:
$\sum_{n=0}^{\infty}c_n(xa)^n$
where $c_n$ are the coefficients of each term in the series and $a$ is number.
To find the Radius of Convergence of a power series, we need to use the ratio test or the root test. Let $A_n=c_n(xa)^n$. Then recall that the ratio test is:
$L=\lim$_{n →$\infty$}$\frac{A_{n+1}}{A_n}$
and the root test is
$L=\lim$_{n →$\infty$}$A_n^{\frac{1}{n}}$
where the convergence happens at $L$< $1$ for both tests. More accurately we can say that the convergence happens when $xa$ < $R$, where is the Radius of Convergence.
The Interval of Convergence is the value of all $x$’s, for which the power series converges. So it is important to also check if the power series converges as well at $xa=R$.

1.
Radius and Interval of Convergence with Power Series Overview

2.
Questions based on Radius of Convergence
Determine the radius of convergence for the following power series: 
3.
Radius of Convergence of Sine and Cosine
Determine the radius of convergence for the following power series: 
4.
Questions based on Interval of Convergence
Determine the interval of convergence for the foll owing power series: