Power Series

Power Series

The series we've been working up to this whole time.

Multiple Choice

 $\sum_{n=1}^{\infty}a\left(r\right)^{\left(n-1\right)}$  What is this series known as?

• Geometric Series

• Alternating Series

•  Telescoping Sum 

• P-Series

Multiple Choice

If I do a partial fraction decomposition on a series that means it's a telescoping series.

• True

• False

Multiple Choice

 $\sum_{n=1}^{\infty}\left(\frac{1}{n}\right)^p$  What is this series known as?

• Alternating Series

• Geometric Series

• Telescoping Sum

• P-Series

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

Radius of Convergence

• 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.

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 ( $-\infty,\ \infty$ )