Checking for Bad Capacitors
Checking for bad capacitors is a skill that is employed in eval steps of computer recycling and refurbishing process, for example eval, build, and quality control. It is not uncommon for machines that have been working fine previously to fail after a period of being powered off (loss of power, moving the machine, or stored for a time).
Look around for capacitors that are leaking or bulging. Check carefully around the motherboard. Video cards and other add-on cards can fail due to bad capacitors, too. You may gently run your fingers over the tops to check if they are flat and relatively flush with their casing, though if doing so try to ground yourself by holding onto a metal part of the case or something else to reduce the risk of damage to the circuit from static, and wash your fingers afterwards.
Check carefully, particularly around the processor/cpu socket (if inspecting a motherboard), as bad caps often look like they do because they degrade quickly, and are therefore likely to not last much longer if still functional at all.
For example images of what burst capacitors look like, check google images or similar for reference.