To Sum it All Up....

The ff arises in an expected value problem in formal political theory.

Let 0<a1,a2,b3,... and 0<b1,b2,b3,... with the a's and b's summing
altogether to 1.

Can the following be summed, or more easily or compactly re-written
(maybe by summing by parts):

(1-a1)b1 + 2(1-b1)a2 + 3(1-a1-a2)b2 + 4(1-b1-b2)a3 + 5(1-a1-a2-a3)b3 +
....

.