Having difficulty figuring out how to determine if a series has a Maclaurin expansion. Anyone think they can clear this up for me?
Well my book defines a Maclaurin series as:
So for example, one of my questions says:
Find the Maclaurin series for f(x) using the definition of a Maclaurin series. (assume that the f has a power series expansion. Do not show that R_n(x)->0)
then they give 6 problems:
If the instructions didn't tell me that I could use a Maclaurin series, then how would I know?
In class, we covered Maclaurin series, then suddenly jumped to vectors (we never got to Taylor series, I think we covered maybe a third of this section). In class, I asked my instructor if every function had a Maclaurin series. (to be honest, I don't really understand how it works, I looked at the graphics for Taylor series such as on Wikipedia, and the graphics make a lot of sense, but I don't really understand how the math relates to the graphic.) My instructor got back to me after the next class, and explained when they do have one, but he had to get to a meeting, and so the explanation was very abbreviated, and I didn't really get it. But my instructor has in the past put problems on the test specifically that were asked to him, such as the integral of sec^3(x). So I am concerned that on the test he will put a question such as "Find the Maclaurin series of f(x)" but the function will not have one. If I am not able to distinguish between functions which do and do not have one, I will lose a lot of points and waste a lot of time. So I figured I'd ask how to determine whether a function has one or not.