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Nominal 25V on the capacitor means it can tolerant up to 25V input I think. What happen to the capacitor is that it store the electricity, not consume, until it is charged up to 3V in the opposite direction to the batteries, meaning that the longer tail of the capacitor becomes 3V. So then the voltage drop between the LED gradually(the speed depends on the capacitance of the capacitor) becomes 0V and the LED.