MemTest86+ is something you can use. Download the bootable ISO image and create a CD from it. Run it overnight at a minimum. You want at least 3 complete passes of all the tests.
Are you overclocking your system? If not then go into the BIOS and select the default BIOS setup and save it. Try again. You may need to change to AHCI again after changing the BIOS.
If your RAM fails then it could be many things causing the problem... The RAM, Power Supply, Motherboard, and possibly the CPU but generally not.
Good Luck