(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(∞∑n=0anxn)⋅(∞∑n=0bnxn)=∞∑n=0(n∑k=0akbn−k)xn
Let us look at some details.
(sum_{n=0}^infty a_nx^n)cdot(sum_{n=0}^infty b_nx^n)(∞∑n=0anxn)⋅(∞∑n=0bnxn)
by writing out the first few terms,
=(a_0+a_1x+a_2x^2+cdots)cdot(b_0+b_1x+b_2x^2+cdots)=(a0+a1x+a2x2+⋯)⋅(b0+b1x+b2x2+⋯)
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=a0b0x0+(a0b1+a1b0)x1+(a0b2+a1b1+a2b0)x2+⋯
by using sigma notation,
=sum_{n=0}^infty(sum_{k=0}^na_kb_{n-k})x^n=∞∑n=0(n∑k=0akbn−k)xn
