More info: MemTest86 is the standard. Put that on a flash drive, boot into it, and run it overnight. It needs to complete a full pass, which takes 4+ hours. A single failure or two is OK, any more is not.
If we are testing hardware, I would also suggest a CPU test with Prime95 an a GPU test with Furmark. Both of these tests are faster than the memory test, and you can always do them from a live linux environment if you want to remove your current installed OS as a factor.