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If the dead section is yellow or brown, then if should be removed as it is dead and cannot be saved. If it is still green, I would leave it be.

If it is a very old plant, sometimes sections can die off due to old age. But, you may want to make sure it does not need repotted and that it has been fertilized.

If you cannot provide enough light for the plant, extra light from a lamps will help. You do not need to invest in a "grow" light. A simple florescent bulb will do just fine. Whether you choose the grow light or florescent light, you will need to make sure that the bulb is placed only a few inches from the plant. Otherwise, the light will not reach the plant well enough to help it.

The following article addresses Euphorbia truncalli (Pencil Cactus): https://www.gardeningknowhow.com/ornamental/cacti-succulents/pencil-cactus/care-pencil-cactus.htm
Yellowing stems can mean several things; age, injury, temperature and too much/little moisture. Sometimes my "pencils' throw off old segments for seemingly no reason. It's kind of like when we lose a hair or two! If the plant has recieved a physical (or mechanical) injury, the connection between stem joints could have been broken. Euphorbia's are notorious for throwing leaves and segments when traumatized by temperatures too low or drought. The plant is trying to conserve it's resources by "downsizing". And, this plant is a succulent, don't keep it wet. Let it dry slightly between waterings.

Euphorbia's are temperature sensative and will drop leaves and/or segments if they get too cold. I lost heat in a greenhouse last winter with temps at 10*F outside. The greenhouse temp dropped to 22 for one night. All euphorbia's dropped leaves, some perished completely, and the 2 big pencils lost a lot of segments. Any night temperature above 55*F. will be sufficient for your plant.
The plant will survive in temperatures down to 40*F., but will not grow. Any legnth of time under 40*F. on a regular basis will leave you with an unhappy camper.

This is typically caused by overwatering. Make sure to let the soil dry out in between waterings. If it was accidentally overwatered in the past, I would check the roots to make sure that there is no root rot.