Question

Expand e^xcosx in powers of x upto x^4​

Answer 1

Answer:

Suppose you have the function:

f(x)=excosx

and you need to find the 3rd degree Taylor Series representation. The way I have been taught to do this is to express each separate function as a power series and multiply as necessary for the 3rd degree. For example for

cosx=∑n=0∞(−1)nx2n(2n)!=1−x22!+x44!+⋯ and ex=∑n=0∞xnn!=1+x+x22!+x33!+⋯

multiply the terms on the right of each until you get the 3rd degree.

Logically, I am happy. However, I have not seen a theorem or any rule that says you can just multiply series in this way. Doing it this way, is there a guarantee that I will always get the power series representation of f(x)?

Additionally, if instead of multiplying, functions were being added? Would the above hold true - take the series of each function and add up the necessary terms?