Can I upgrade an i3 CPU with an i5 CPU on a gateway SX2840-01?

Your gateway PC have 1st genenration Core i3 processor and first generation LGA is 1156 so if you want to replace your processor you replace with first generation core i5 or core i7 and my advice is not replace i3 to i5 go to i7 because first generation i7 still cheap and good and you can enjoy latest game like fallout and other games

It uses a i3-530 which is a first generation i3 CPU. The only i5 or i7 CPU's which would work would be first generation. The best I can tell that computer uses an Intel H57 chipset. Below is a list of CPU's which should be compatible. Whether Gateway did something to the BIOS to prevent higher end chips from working I could not say. This is a list of CPU's officially supported by the H57 chipset. Also linked below is the motherboard reputed to be for your model. The K series CPU's overclocking function likely will not work.

I would note that only first generation Core i7 Processors and 700 series i5 Processors are quad core. The 600 series are dual core and not much of an upgrade.

Overall it would be a bad idea to upgrade this as a gaming computer. The H57 chipset is limited to 8 PCI-e 2.0 lanes. That PCI-e x16 slot is likely maxed out at PCI-e 2.0 x4 as 1 lane is used for the PCI-e x1 slot and other lanes are needed for other components. The case is also small form factor so you'll need a new case for any decent video card.

I suppose you could load it up with used parts I wouldn't waste money on new parts I'd mostly find used ones for cheap on eBay. Toss in a Core i7-870, 8GB (4x2GB not 2x4GB) RAM and a GeForce GTX 670 2GB. Then get a cheap Micro ATX case from a decent brand like Coolermaster or Thermaltake and a Corsair CX500 PSU. It'll be much faster than a new system you could build for that price

If y spend money everything is possible