A DA40 pilot not in communication with ATC reported a NMAC when a regional jet airliner climbed towards them on a converging course. The reporter was aware of the traffic on their flight display and was monitoring the ATC frequency but took no action to avoid the conflict.

Narrative

I was flying VFR above the ZZZ class C airspace at 5500 ft. I was monitoring ZZZ Approach on XXX.X but was not receiving services due to radio congestion and I would be staying clear of the class C airspace below. I saw Aircraft Y turning towards me on my EFB and G1000 flight display as I was transitioning over the ZZZ class C airspace; just northeast of ZZZ1 airport. Initially Aircraft Y was well below me; however the Controller cleared them to climb and I became concerned; as their trajectory would conflict with my flight path. Indeed; just a few moments later; their turn continued directly into my path. Their rapid speed (nearly 300 kts.) and vertical speed brought them dangerously close to my path at 5500 ft. as I was passing just northeast of ZZZ1 airport. I turned back when my G1000 issued a traffic alert and clearly saw Aircraft Y converging. I could not out climb this E175 so I stayed level at 5500. The pilot of Aircraft Y notified ZZZ Approach that they received a TCAS RA and would be descending. They descended below me and continued their turn before climbing back on course. Had Aircraft Y not taken evasive action; they would have most certainly crashed into me. They did not seem to have visual contact; but they were converging on me from below and behind. ZZZ Approach did not seem to notice the potential conflict. They did not call me out to the pilot of Aircraft Y; despite my plane being equipped with ADS-B and enhanced surveillance; which should have indicated my selected altitude of 5500 ft. as well as my selected heading. ZZZ Approach also cleared Aircraft Y to continue their climb despite this potential conflict; further support for the fact that they must not have noticed me on their radar scope. Utilizing ADS-B data I was able to verify that my ADS-B and enhanced surveillance were working as expected. I also used this data to recreate the flight path for both my plane and Aircraft Y.