Hey Nappy.

Assuming I.I.D for the e(t) processes, you can use Var[aX + bY] = a^2Var[X] + b^2Var[Y] where a and b are non-deterministic quantities not related to the X's or Y's in any way (completely independent).

Since you know the sequence is summable, then it should be part of l^2 which means that the inner product of <x,x> (x is your sequence represented as an infinite vector) should also be finite (summable) so the sum of variances should be summable.