Download and create a Memtest86+ boot disk or CD
See:
http://www.memtest.org/
Turn off the PC, remove the AC power cord.
Wait 5 minutes or until the green motherboard LED (if there is one) goes
out.
Now remove the 512MB stick, insert the 1GB stick.
Power up the PC and running Memtest86+,
this runs from the boot disk or CD you created and
should eliminate or confirm if your ram is bad.
Let it run for as long as you can, 2,4,6,8 or more hours,
if no errors by then your ram is OK.
Again turn off the PC, remove the AC power cord.
and wait 5 minutes or until the green motherboard LED goes out.
Insert the 512MB stick so you now have both sticks installed.
Run Memtest86+ again.
Last test is Prime95
http://www.mersenne.org/freesoft/
It's a stand alone .exe file contained in a .zip archive.
Just choose to run the "stress test" option for 3 hours.
If your PC can pass this test both your memory and CPU
are fine (close the case cover to maintain proper ventilation)