He's taking the running average of the partial sums:

    1-1 = 0
    0+1 = 1
    1-1 = 0
    0+1 = etc etc

The partial sum would end up being equal to the final sum by definition when you're done summing all items. Since we're talking about infinite sequences you're never done summing all items so you'll have to do something else to end up with an answer. for example seeing which way the partial sum trends. In this case it trends solidly in the direction of 1/2