What does $a$ mean in Taylor series formula? I'm trying to code the Taylor summation in MATLAB, being Taylor's formula the following:  

I've also seen $a$ denoted as $x_0$ in distinct bibliography.
Problem is that I'm not sure how should I evaluate or assign for $a$.
At lecture I studied the exponential Taylor's representation being: 
And I got to that same summation by evaluating $a$ as 0 but and I know that is meant to be McLaurin formula, but what's within evaluating with a different value or letting $a=x$?
What should I do in order to code properly Taylor summation for distinct functions?

$a$ is any complex$^*$ quantity you want, so long as it is independent of $x$ and $n$. If you're writing a generic "evaluate a Taylor polynomial" function, you'll probably want to accept $a$ as input from a user rather than supply a value yourself.
There are two typical reasons to pick any particular value of $a$:


*

*Your choice of $a$ makes it easy to obtain the values $f^{(n)}(a)$, and/or it simplifies those values

*You want to evaluate it at $x$'s in some range, and you choose $a$ to keep the value $x-a$ small over that range, so that most of the value of the function comes from the first few terms of the sum


*: Actually, you can even do things in more generality than the complexes!
