What is the best way to join the stair rail end piece to the stair rail. I have both the stair rail and the end piece cut at 45 degrees. The end piece is 3 1/4 inches long. Thanks for all suggestions.

You have the right idea. 45 degree joint, drill a dowelplug hole(make sure you have the dowelplug size 1st e.g. 3/8",1/2"), and predrill a pilot hole underneath rail prior to assembly.. Use plenty of glue and woodscrew it together. Off set the pieces to allow the screw to pull it into position. You see what I mean when you try it.
It's OK , you can reposition it. When it's together , let the glue set, use filler and sand in the profile of the rail to match. Install a dowel plug underneath to hide the screwhead.