Also check the cable crimps (earth and HV feed), if coated in fluffy muck, change ASAP, but in the short term clean with bicarb mixed with boiling water, then when nice and clean pound the crimp closed with a hammer or crush in a vise. good solid contacts are a must.
Your local motor factors will probably do a cold crank test on your battery.
Give it a full charge, remove it from the Jeep, wait a few hours then go and ask them to test it.
Or put battery on charge, but disconnected from the Jeep. let it cool for a few hours afterwards and then check what the no load voltage is, 12volts is marginal (12.8 to 13.2 much better) anything in the 10V area and its stuffed.
ALWAYS wear full protection when working with the battery as it is far too easy to make mistakes where it gets shorted out and can sometimes burn or boil up all over you.
They look innocuous enough but they can be really mean.
OR (Only if you are confident).
If you are able to loosen the plugs on the top of the battery cells, (ie it's not sealed for life). Then the way we oldies used to test them was to take the HT leads off (so engine wont start), loosen but NOT remove the caps on the battery and crank the engine. Watch for 1 cell bubling away more than the others, the cap jiggles about as the acid gasses under heavy load, if this happens it's a pretty sure bet the battery is history, but if you are at all unsure, then DONT try this test.
