Trackpad failing to connect

iMac, OS 10.11.6, magic trackpad:

My trackpad refuses to connect successfully - as far as the bluetooth preference pane is concerned, it's cycling between connecting and disconnecting every 10 seconds over and over, but even when it says it's connected it's not useable.

I've tried turning off/on again both bluetooth and the trackpad, changing batteries, and I've tried removing the trackpad from the bluetooth list and redoing pairing from scratch, none of it works.