It's simple mathematics ..
For sake of simplicity ,
Let f(x) = a0 + a1 x + a2 x2 + .....+ ak-1 xk-1 + ak xk .......
g(x) = b0 + b1 x + b3 x2 +.....+ bk-1 xk-1 + bk xk ......
So if we consider the coefficient of xk in f(x) . g(x) ,
Then we have ( a0.bk + a1.bk-1 + ................ + ak.b0 ) ..............(1)
So general term of above summation (as shown by (1)) can be wriiten as : aj . bk-j where j varies from 0 to k..
So it can be written as : Σ aj . bk-j where j varies from 0 to k.
Likewise we do for every xk where k starts from 0 and goes to infinity..
Hence f(x) . g(x) will contain all the terms of 'x' having exponent from 0 to infinity
which is nothing but = Σ ( Σ aj. bk-j ) xk where the limits of inner summation is from 0 to k as explained above and limits of outer summation is 0 to infinity..