Good troubleshooting!
From what I've seen on the 17 inch R3 and R4's, the 8 beeps is indicative of the GPU, not the monitor like Dell will tell you.
You can find the part for less than half that on Amazon and even cheaper on the big nasty auction site. Often I have purchased a whole broken laptop cheaper than the individual part(s) I needed.
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Gollnick is correct: You can use a charger if output voltage is within +/- 5% of the original charger's rated output. So a 19.5 V will let you get away with 18.5 to 20.5 Volts. Polarity is important as is the fit of the adapter plug. You don't want it arcing internally or sliding out easily, nor do you want to have to cram it in place thereby stressing the internal connector.
Dell power supplies are almost universal in voltage at 19.5 volts. The amperage / wattage varies with use. (Amps x Voltage = Watts)
The 240 Watt supply (and therefore higher amperage) is for the increased "performance type" parts in the higher end XPS and Alienware gaming laptops. 120Hz 3D ready high definition screen, discrete graphics (sometimes dual, although not on the R3), 10-15K RPM hard drives (sometimes dual), multi-core processors (sometimes dual and / or desktop processor), and even dual batteries (but I don't recall them on this model). Wattage / amperage is critical on these systems. You can have all the voltage you want but if you don't have enough pressure behind it pushing it in fast enough, you get problems. Use an adapter with too low a wattage rating (like a 65 watt), and the battery won't charge while the system is running. I run into this with customers all the time in their vehicles. They purchase a little 45 watt power supply for their cigarette lighter / accessory outlet and their laptops just don't charge (or even boot) while on the road.
Try the original power supply in the unit again. If you only have the smaller adapter, pull peripherals like the hard drive, battery, and DVD ROM drive and see if it will at least boot into BIOS for you. If not, then we are probably back to the bad GPU.