I was wondering if anyone knew rigorous proofs from the following properties of Taylor series:
(a) Addition and subtraction of series, (eg: taylor series for sinx+cosx = taylor series for sinx + taylor series for cosx )
(b) Substitution of z for something else to form a new Taylor series for a different function, (eg: replacing x with x^3 in the taylor series for e^x to give taylor series for e^(x^3) )
(c) Multiplication and division of series. (eg: taylor series for sinx*cosx = taylor series for sinx multiplied by taylor series for cos x )
I am a Mathematics undergraduate and am thus very familiar with using these properties, but I have never seen a rigorous proof for any of them.
If anyone could post any of the proofs then I would be extatic!
Many thanks in advance.