Maclaurin/Taylor/Power Series clarification

Am I right in thinking that a Maclaurin or less specifically Taylor series is the infinite sum representation of a function?

So If I want to calculate e^5 exactly and I had infinite time and patience, that If I calculated out:

$\displaystyle 1+5+\frac{5^2}{2!}+\frac{5^3}{3!}+...$ That I would arrive at the value for e^5?

If so, then I've seen it works with trig functions, log functions and x to the power of some changing variable. It also works for fractions of all of those... Are there any limitations? Or can ANY function be represented as a Maclaurin or Taylor series