Someone here said the problem was probably a leaking capacitor. I took the
cover off, but the capacitor looked fine. But when I touched the capacitor
the opener suddenly started running. It was okay for 2 days, then stopped
again. I touched the capacitor again and it ran okay for 5 minutes, then
stopped. Now touching does not help.

I figured there was loose wire, but I sure can't find one. Liftmaster says
it is probably a bad capacitor for $20; and if not that, a bad logic board
for $85.

Does a bad capacitor make sense? Could touching a bad capacitor make it run
for a while? Obviously they won't be taking the capacitor back if it
doesn't help.
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I'm can't help you troubleshoot the problem, but if you want to try a
new capacitor, take the old one along with the wiring schematic to an
electronics store. You can get a new one for under a dollar.