Hi everyone!

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.

Radders.