How to multiply these summations ?

[praecox]

Ok. I feel a little dumb for asking, but I'm working on a abstract algebra proof and this has got me stuck:

\left(\sum_{i=1}^{n}a_ib_i\right)\left(\sum_{i=1}^{k}a_ib_i\right) where n,k are some positive integers.

I feel certain that it's not just a sum to n+k or nk, but I could be wrong. any help would be awesome. :)

 

[aesir]

It can be written as a single sum with the cauchy product.
The terms of the new series are the convolution of the original terms.

An elementary way to see it is: attach x^i to each coefficient, then group in a single series in powers of x, then put x=1.
Like you would do for (1 + x + 3 x^2)(2 + x + x^2)= (1+2) + (1*2+1*1)x + (3*2+1*1+1*1)x^2 + ...