How do you find the product of power series?

#(sum_{n=0}^infty a_nx^n)cdot(sum_{n=0}^infty b_nx^n)=sum_{n=0}^infty(sum_{k=0}^na_kb_{n-k})x^n#

Let us look at some details.

#(sum_{n=0}^infty a_nx^n)cdot(sum_{n=0}^infty b_nx^n)#

by writing out the first few terms,

#=(a_0+a_1x+a_2x^2+cdots)cdot(b_0+b_1x+b_2x^2+cdots)#

by collecting the like terms,

#=a_0b_0x^0+(a_0b_1+a_1b_0)x^1+(a_0b_2+a_1b_1+a_2b_0)x^2+cdots#

by using sigma notation,

#=sum_{n=0}^infty(sum_{k=0}^na_kb_{n-k})x^n#