Convergence. We've achieved it. PC monitors can do anything TV sets can do, other than directly tune TV channels. And TV sets generally have a VGA or RGB mode that allows them to operate as well as any PC monitor.
About the only real differences are that TV sets are more versatile and have more inputs and adjustment parameters, while PC monitors seem to have more choices for matching existing arcane display rates and pixel map sizes common to the PC world.
PC monitors used to be either the older 4:3 versions, or newer 16:10 versions (at work I actually have a Magnavox HDTV [I did not choose it] that has a 16:10 PC display, making everyone look like they've been on the Atkins diet). Anything other than the 4:3 version will stretch whatever is on your PC, but this is not really problematic the way 16:9 HDTVs can make every hot chick look like she needs to drop 30.
Until a year or so ago all wide-screen PC monitors were 16:10. Now, most of them are 16:9. I would like to think that PC makers have given over to HD aspect ratios as a nod to convergence, but I think it is probably due to the realities of the marketplace, where 16:9 displays have become much cheaper and commoner than 16:10 displays, and so they go the cheap route. OK, no harm, no foul.
But I will never buy a PC monitor again. In fact, my current PC monitor is actually a 22" Vizio TV in RGB mode handed down when I upgraded a bedroom TV. Works like a champ.