Power Series

What?

It’s a way of representing a function, centered around a given point, by a Series. It’s in the form

$f(x)={\sum }_{n=0}^{\infty }{a}_n(x-c{)}^n$

An example of $f(x)=e^x$, where $a_n=\frac{1}{n!}$ and $c=0$.

$$f(x)=e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}$$

How do you convert / find a Power Series?:

Check Converting Function to Power Series

Differentiation / Integration:

Think about it. If they perfectly represent a continuous function, then they’re differentiable. We simply differentiate / integrate each term.

$f^{\prime }(x)={\sum }_{n=1}^{\infty }n{c}_n(x-a{)}^{n-1}=c-1+2c_2(x-a)+3c_3(x-a{)}^2+\dots$

Relation to Taylor Series:

Taylor series are simply a specific type of Power Series, except they’re about using derivatives at a point to gain information about the function around that point.

What happens if you extend the Series out into infinity?

Well, if it only ever gets better and better at representing it, then it’s known to converge. Otherwise, if it hits a point on either side of the original point that it oscillates around, then it’s known to diverge. The points on either side are known as the Radius of Convergence.