Let’s count. Sum and product of formal power series

Do your best in trying to solve the following problems. It is normal that this step will take some time, we recommend to perseverate and wait before looking at the complete solution of the exercises in the video or in the pdf below.

Exercise Compute (A(X)+B(X)) and (A(X)cdot B(X)) for [A(X)=2+2X+2X^2+2X^3+cdots=sum_{n=0}^infty 2X^n,] [B(X)=X+2X^2+3X^3+cdots=sum_{n=1}^infty nX^n.]