Usually it is done the other way around given the Taylor Series of a function what does a change in the function do to the series.
For Eg e^x = sum (x^k/(k!))
e^(2x) =sum [(2x)^k/k!] = sum(2^k) (x^k)/k!)
Or 1/(1-x) = sum (x^k)
1/(1+x) = sum ((-x)^k) = sum ((-1)^k x^k)