A capacitor does not and cannot work as a 'power saver' or, more-accurately, 'energy saver'.
However, a capacitor can be used to reduce the power factor of a load. Reducing power factor, though, has absolutely no effect whatsoever on the amount of energy consumed by that load and will not reduce the energy bill for a residential consumer such as you or me.
What reducing power factor does is to reduce the load current which, in turn, will reduce the amount of copper that must be provided by the utility company, in terms of its cables and Transformers. So power factor 'improvement' can save the utility company money, by allowing it to use smaller cables and transformers to supply any given load.
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When DC power is first applied to an uncharged capacitor it appears as a short circuit.
Because a capacitor is ideal for storing energy over short periods, as in a reservoir capacitor in a power supply.
In a battery or capacitor.
Assuming the question refers to a capacitor-input power supply with a transformer, rectifier and reservoir capacitor, the problem is caused by the high peak currents which flow for only a part of the ac cycle while the capacitor is charging. Thus the poor power factor is not caused by the usual difference in phase between the voltage and the current, which is easily corrected. This being the case the power-factor can't easily be corrected and this type of power supply is confined to low-power applications of less than 1 kW.