Take a light bulb, and carefully break the glass, making sure not to damage the filament inside. When you remove the glass, you release the argon inside and expose the filament to oxygen. This means that the next time you turn the bulb on, it will burn out almost instantly.
Place an unplugged lamp in position for the photograph. Carefully screw in what's left of the light bulb.
Set up your camera in position to take a series of high speed shots.
Plug a power strip into the wall near the camera. With the lamp still unplugged, turn on the lamp switch.
Stand ready with the lamp plug at the power strip, and then activate the camera. Immediately plug in the lamp. If you timed it right, one of the photos you took will have captured the filament lighting up before it burns out. Unplug the lamp.