if you multiply two summation notation
ex.
(summation (x)) *(summation(y)) do you get (sumation (xy)) ??
So try :
$\displaystyle \left(\sum_{i=1}^2 a_i \right) \times \left(\sum_{i=1}^2 b_i \right)=(a_1+a_2) \times (b_1+b_2)$
$\displaystyle \sum_{i=1}^2 a_i b_i=a_1b_1+a_2b_2$
is $\displaystyle (a_1+a_2) \times (b_1+b_2)$ equal to $\displaystyle a_1b_1+a_2b_2$ ?
And hence is your formula correct ?
You can see here : Cauchy product - Wikipedia, the free encyclopedia for the formula of the product of two summations