Charge up two large caps with 10v each. Connect them in series. What do you get across the two? You get 20v because the cap voltages add.
The cap is like a battery for a time, so replace the first cap on the left with charged battery and go from there for that next half cycle.
But it depends on the polarity. If you connect two charged batteries in series with opposite polarity, you get ZERO voltage (assuming equal voltages). In other words, YES, the voltages add, but if you add x and -x, you get 0. That's my problem here.