OK I understand about the problem with HP needing to want to fix any problems with their products.
The reason that I asked if there is any warranty on the notebook was because If there is HP would normally be liable for its repair, opening up the case yourself gives any manufacturer enough of a reason to not honour any warranty that there may be.
If there is no warranty I would still go the local tech route first, they could check both the RAM and power issue in ten minutes and may not even charge a fee, if neither are the cause of the issue you have nothing to lose by investigating further yourself.
Asked previously but you may have missed it, does the internal cooling fan spin up at all?
If you do not hear the cooling fan running and the base of the notebook gets hot it suggests that the cooling fan is not working at all, you would normally get an onscreen thermal shutdown warning when this happens and the fact that you do not could mean that something may have burnt out, this does mean that the notebook needs to be opened up and the CPU cooling fan tested/replaced.
Edit to add;
Scrap the above Marty, after going over the disassembly steps it appears that your model of notebook does not have an internal cooling fan, the fact that the notebook is getting warm is normally a good sign in that the MB and CPU are getting power, if either component was bad the notebook would be cold, back to getting the RAM checked out I`m afraid.
Do you have access to a screen that has a HDMI port that you could try hooking the notebook up to and see if you get any display on that, wanting to make sure that we check as much as we can here.
I really would not purchase anything before getting the notebook checked out by a tech (if you need to) RAM has come down in price but is still costly.
Seeing if you are able to borrow a compatible stick of RAM for testing purposes and having the notebook and existing RAM tested by a local tech/store are all that I can suggest I`m afraid, purchasing a new stick of RAM is not a financially viable option atm.
Edit to add: Have you been able to try connecting the notebook to a HDMI screen as was suggested, if yes, what was the outcome.
