Power Series

 $\sum_{n=0}^{\infty}c_n\left(x-a\right)^n=c_0+c_1\left(x-a\right)+c_2\left(x-a\right)^2+...$ 

There are only 3 possibilities for convergence:

Converges only when x=a

Converges for all x

There is a positive number R such that the series converges if |x-a|<R and diverges for |x-a|>R

When x=a, R=0

When the power series converges for all x, R= $\infty$  

Bessel Functions are a great example of a Power Series....Some of them can explain the temperature distribution on a plate or the vibration a circular drum head.

We're going to walk through how to find the radius of convergence by using the ratio test.

 $\sum_{n=0}^{\infty}a_n$  is absolutely convergent when  $\lim_{n\rightarrow\infty}\left|a_{\frac{\left(n+1\right)}{a_n}}\right|=L<1$  

Find the radius of convergence of  $\sum_{n=0}^{\infty}\frac{\left(-1\right)^nx^{2n}}{2^{2n}\left(n!\right)^2}$  using the ratio test.

 $\left|\frac{a_{\left(n+1\right)}}{a_n}\right|=\left|\frac{\left(-1\right)^{\left(n+1\right)}x^{2\left(n+1\right)}}{2^{2\left(n+1\right)}\left[\left(n+1\right)!\right]^2}\cdot\frac{2^{2n}\left(n!\right)^2}{\left(-1\right)^nx^{2n}}\right|$  

At this point, it's an algebraic simplification.

As n $\rightarrow$  $\infty$ , the limit approaches 0<1 for all x, so the Radius of convergence is ( $-\infty,\ \infty$ )