A power series is sort of like a polynomial with an infinite degree.

A power series is of the form:

$$ \sum_{n=0}^\infty c_nx^n $$

Here, $x$ is just a variable that we can evaluate at different numbers.

The $c_n$ values are constants called the coefficients of the power series.

It’s often useful to pretend that power series are functions of $x$, (i.e. $f(x) = \displaystyle\sum_{n=0}^\infty c_nx^n$).

A power series of the form $\displaystyle\sum_{n=0}^\infty c_n(x-a)^n$ where $x$ is a variable, $c_n$ are constants, and $a$ is constant is called a power series centered at $a$.

examples

$$ \sum_{n=0}^\infty x^n = \frac{1}{1-x} = f(x) $$

This equality is only true for $|x|<1$.

$$ \ln(x) = \sum_{n=1}^\infty(-1)^{n-1}\frac{(x-1)^n}{n} $$

This is true for $x$ close enough to 1.

why are power series useful?

Power series are useful because they allow us to treat many functions as if they were power series, which are easy to approximate.

an important note

The expression $0^0$ is an indeterminate form, and for that reason it’s sometimes left undefined. However, in power series, it’s more useful to define $0^0 = 1$.

intervals/radius of convergence

For $\displaystyle\sum_{n=0}^\infty c_n(x-a)^n$, exactly one of the following is true:

1. The series diverges for every $x\ne a$.
   1. Example: $\displaystyle\sum_{n=0}^\infty n!x^n$ (here, $c_n = n!$)
2. The series converges for every value of $x$.
   1. Example: $\displaystyle\sum_{n=0}^\infty \frac{x^n}{n!}$ (here, $c_n = \frac{1}{n!}$)