Jason Rasmussen (student)

Thank you, for the response.

I suppose my confusion comes into play when I am trying to figure out where the xⁿ term comes from. I know that the Power Series notation is directly related to the Geometric Series of the form sigma [ brⁿ ] where the limit is b/(1-r) for convergence at | r | <1. Therefore, the function f(x) needs to somehow take the form of b/(1-(x-a)), which may take some manipulation, and by setting r = (x-a) and b = C_n, we get the Geometric Series converted to the Power Series. Taking the nth order derivative of the Power Series gives C_n = fⁿ(a)/n!. There must be a gap in my knowledge somewhere because I cannot seem to make f(x) = e^x take the form of f(x) = b/(1-(x-a)). Maybe I should have labelled my question as "middle" because it may be more of a personal problem with algebra and logarithms. Or, am I to assume that all functions can be represented by sigma [fⁿ(a) * (x-a)ⁿ / n!]?

The usual way to approach Maclaurin series is to assume that you have a function f(x) that can be represented by a power series

and then show that C_n must satisfy

This would be a geometric series if each C_n had the same value, but that us usually not the case.

The answer to your question "am I to assume that all functions can be represented by sigma [fⁿ(a) * (x-a)ⁿ / n!]?" is no.