Should I just memorize the common Taylor and Maclaurin series like the sinx e^x ln(1+x) or should I really understand how it works?
It night help to read on Taylor’s theorem:
http://en.m.wikipedia.org/wiki/Taylor%27s_theorem
Given that the Maclaurin series of e^x, sin x, cos x, etc. are just applications of Taylor’s theorem at x = 0, you can just memorize these.
I think you should mainly understand how they work as some problems which require you to come up with terms of a taylor series that are NOT the common ones.
Ex: They’ll give you a random summation and ask you to expand it.
Memorizing the common ones like the above poster listed are helpful for multiple choice as you can just use substitution to derive a lot of other Taylor Series which can obviously save a lot of time, but you ultimately need to understand them.
It doesn’t take that long to memorize it. You should know how to derive it just in case you forget the summations though. There is a high chance that it will pop up in the FR of the noncalc, so be prepared for it! Sometimes, it may be a similar summation and just altered with few numbers (like maybe e^(3x)).
Also, Maclaurin series for sin(x) and cos(x) should *definitely * not be mixed up. Otherwise, a kitten dies.