Here we will look at the most typical situations one can encounter when putting together functions and series using operations. Taking it from the easiest, one may face the following problems:
    a) functions are added/subtracted; this is simple, you just have to make sure that the series have the same powers and the same indexing, see Manipulating (power) series in Methods Survey - Series of functions.
    b) a function gets multiplied /divided by some term of the form (x − a)ⁿ. Then you simply multiply/divide all terms of the series; you just have to be careful that dividing makes sense (a has to be a root of that function with sufficient multiplicity).
    c) functions are multiplied; Cauchy product of series is possible theoretically, but it usually does not lead to a nice answer. Good luck.
    d) functions are divided; dividing series by a series is tricky at best and it is done using the method of undetermined coefficients. Unless a miracle happens, you do not get a generic formula for terms and therefore you won't get a formula for the whole series; in a typical case you only get to know its first several terms (but you can find as many as you need).