I'm trying to install a Radeon HD 3650 256MB that I bought used from Amazon in my HP a1640n desktop (Windows XP Pro SP3 32-bit). However, whenever the card is in the machine, it can turn on, but only to a black screen (no signal). No BIOS screen, no sounds, nothing. It was the same whether the monitor was plugged into the motherboard VGA port or the card DVI port (using an active DVI to VGA cable). I even tried going into the BIOS to set the primary display device to PCIe instead of PCI as well as trying to boot into Safe Mode after installing the card to disable the Intel integrated graphics drivers. I couldn't get to Safe Mode with the card in the machine due to the black screen problem. Is the graphics card or DVI-to-VGA cable broken, or is there something obvious I'm missing? I don't believe I have any other equipment to test the card or cable.
Your problem is probably an insufficient power supply.
ATI Radeon HD 3600 Series System Requirements include a "400 Watt & 22 Amps" power supply.
The power supply of the HP a1640n is apparently a big secret, since HP will not divulge it. I found the specs for the Power Supply Replacement which say "300 Watt Replacement Power Supply", so might be insufficient for this video card.
You might therefore try to upgrade the PSU.