Lighting – Why does 36W LED bulb not light up

First of all, when you read about a "50W LED", it is certainly saying that it outputs the equivalent light of a 50W incandescent light (although that is often a exaggeration). That "50W" LED actually draws less than 10 watts. This means that it draws much less current and emits much less heat, both of which are the factors that typically limit the allowable light in a fixture.

You'd be able to replace a 50/100/150W 3-way with a lower wattage, even if staying with incandescent.

Second, unless the LED light is marked "dimmable", don't try to use it on a dimmer. You may damage the dimmer, light, or both. Even if the light is dimmable, some dimmers will not work well with a low load: a dimmer that works on a 100W light bulb may not work on an equivalent-light-output 13W LED. They make LED/CFL specific dimmers that may work better.

As for the 130V R40, that is probably a "long life" bulb. Running an incandescent at lower than rated voltage extends its life. Therefore, they make 130V bulbs to run longer on 120V. Just use LEDs rated for 120V.

As for your existing florescent lighting, I assume you're talking about the straight tube type. For a fixture that old, it's probably less efficient that newer florescent fixtures or LEDs but still much more efficient than incandescent. This is harder to replace with LEDs, just due to the area covered. There are LED tubes made to replace florescent tubes but they are still expensive and generally require re-wiring the fixture. These are most often used in commercial applications.

I found the answer! @diceless set me on the right track in looking at where the foot of the bulb is connecting to the fixture.

Since it was an older fixture (installed in the 80's, I think), the hot contact (Number 1 in this diagram ) was bent down, and it was connecting with the smaller bulb but not the bigger LED bulb.

With the circuit turned off at the breaker I reached in with a wooden tongue depressor and bent it upwards slightly. Then, I made sure to screw the LED bulb in tightly. Flicking the switch it was bright as day!