Drywall Question

[WeezyBengal]

I asked this in another thread but maybe someone can help me out if I start a new one:

So I basically have about a quarter inch gap between my drywall and finished tile. I had to cut the drywall to install new backer board and i was left with a gap once the tile was finished.

So my question is, how do I feel this in so its all flush? Can I just fill the hole with joint compound? Do I have to use tape? Im not really joining two pieces of drywall since its being filled up against finished tile...

Any recommendations would be helpful.