Power series and Taylor series are ways of showing functions as long sums of numbers. Even though they look similar, they have different uses and important differences.

What They Are:

• A power series looks like this:
  $\sum_{n=0}^{\infty} a_n (x - c)^n$
  Here, $c$ is where we focus on, and $a_n$ are numbers that can change.

• A Taylor series is a special kind of power series. It tries to closely match a function $f(x)$ at a specific spot $c$ using something called derivatives. It is written like this:
  $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!} (x - c)^n$

How They Are Used:

• Power series can describe functions in a certain range, but the numbers ($a_n$) do not need to be related to the functions' derivatives.

• Taylor series are used when we want a good guess of what a function looks like. This is useful when we need to find out what the function equals or when we want to make complex problems simpler.

When They Work:

• Both series have areas where they work well, but the rules for each might be different. A power series works when this is true:
  $|x - c| < R$
  Here, $R$ is called the radius of convergence.

• A Taylor series will match the function $f(x)$ under certain rules. If those rules are met, it gives a complete picture of the function.

Knowing these differences is important in advanced math. It helps us use tests for finding out if series converges, like the Ratio Test or Root Test.