Taylor and Maclaurin Series

The question of taylor and Maclaurins series can start with asking the question about which functions have power series representations and how we go about finding them.

Suppose that we have some function $f$ that can be represented by a power series.

$f(x) = c_0 + c_1(x-a) + c_2(x-a)^2 + c_3(x-a)^4 + .... |x-a| < R$

Now we can start with the $x = a$. It doesnt have to be a, but we're using it in this case. We could also have the formula like this:

$f(x) = c_0 + c_1(x-1) + c_2(x-1)^2 + c_3(x-1)^4 + .... |x-1| < R$

Here, $a$ is swapped out with $1$. So now if the value of $x = 1$, then we are multipling the constant $c_i$ by $0$ because $1-1 = 0$.

The end part of that equation is mentioning details about the radiance of convergence. Which I'll go in to briefly.

Radiance of convergence.

When you're dealing with a power series like the one above there are different ways that the series can converge. The power seris will always converge at it's center. But it can converge in other ways too. There are three ways that it can converge.

1. The series converges only when $x = a$
2. It converges for every value of $x$
3. It converges for some positive number R. Where R > |x-a|