I don't think that because a computer uses 1000W of power it would produce 1000W of heat though. I think if that were true then there could be no computation done since all the energy of the electrons would be converted to heat. I'm not sure what the efficiency rating of computers is, or how much is turned heat.
Not to rain on your back of the napkin math. Although I think your sentiment is right, the cooling systems for computers in space would need to be bigger since you can use convection to move the heat away.
A gaming computer can use 1000W. Google says spacecraft radiators reject 350W/m^2.
If my math is correct (no guarantees) you'd need a king-sized-bed-sized panel just to cool your rig.