E. Taylor Series

Some functions can be written as an infinite polynomial over a specific interval (i.e., $\sin x, \cos x, e^x$)

We refer to this infinite polynomial as the power series of the function and the interval over which the power series converges and is typically equal to the function is the interval of convergence
- This interval is symmetric (except for the endpoints), and is centered at a value $a$
- Radius of this interval is the radius of convergence
Describing a function by a power series allows us to estimate a potentially complex function by a polynoial of any desired degree
We can use the power series of a function to compute limits involving that function, solve ODEs involving that function, and estimate integrals involving that function
We learned two methods for finding a power series for a function $f(x)$:
1. By finding the general form of the $n$th derivative of $f(x)$ and using the ratio test to find the radius of convergence
2. By manipulating the power series that we already know to build power series for new functions
  - In this case, we need to keep track of the radius of convergence as we manipulate the series