Coefficient of the product of two power series

[hholzer]

If a_0 + (a_1)x + (a_2)x^2 + ...
and
b_0 + (b_1)x + (b_2)x^2 + ...

are two power series and the coefficient
of x^r from their product is a power series:
(a_0)(b_r) + (a_1)(b_(r-1)) + ...

What principle or theorem or definition(s)
are we applying when finding that this is
indeed the coefficient of the x^r term of
the product of two power series?

 

[disregardthat]

Well, you will get that $$(a_0+a_1x+...)(b_0+b_1x+...)=(\sum_i a_ix^i)(\sum_i b_jx^j)=\sum_{i,j}a_ix^ib_jx^j=\sum_{i,j}a_ib_j x^{i+j}$$ since every term is multiplied with every other term in the other factor. However you want to extract the coefficient in front of x^k, and that is by looking at the expression above obviously $\sum_{i+j=k}a_ib_j=\sum_{i=0}^ka_ib_{k-i}=a_0b_k+a_1b_{k-1}+...+a_kb_0$

 

[tiny-tim]

hi hholzer!

What principle or theorem or definition(s)
are we applying when finding that this is
indeed the coefficient of the x^r term of
the product of two power series?

i don't think it has a name …

it's just obvious ​