|Try unplugging the tower and then opening up the case. You should exercise static discharge precautions to protect the sensitive components. Ground yourself. If you have 3 prong wiring the case to your computer should be grounded, even with the power strip in the off position. Simply touch the case before touching anything inside the case.|
Now, I suggest you re-seat everything inside the case EXCEPT the CPU and heatsink/fan. Unplug wiring, one by one so you don't lose track of where they go. remove all add in cards if present and reinsert them fully.
Remove the memory sticks, one by one and snap each of them in and out 4 or 5 times to burnish the contacts.
While you have the case open I suggest you also blow the insides out using canned or compressed air. This will be messy so you may want to move the tower to a better location. Pay close attention to the power supply. Blow it out from both ends.
While you have the case open, examine the capacitors for bulging or leaking. See the link below to know what to look for.
One short beep is normal when starting the computer. Are your beeps long or short? Either way, beeps can be grouped with multiple beeps in a group with a slight pause between then a repeat of the original group.
When booting the computer you may see a message telling your what BIOS brand a version you have. You can use that to determine what the beeps mean. The BIOS chip is also labeled. AWARD and AMI are the most common.
The cables to the external hardware can also corrode so re-seating them is also good. The monitor cable should be fastened using the thumb screws to keep it tightly seated.
Of course you need to eventually replace that CMOS battery.
Once the computer is running, I suggest you download and run in real time a utility called HWMonitor. That program can show you what the temperatures and voltages are at any time. A failing power supply could cause problems. The main voltages are 3.3V, 5V and +12V. All those should run +-5% of those values.
Get HWMonitor from the link below.
You might also check the condition of the memory using Memtest86+. GEt that from the link below.
Hope this helps.