Forget it, I'm an idiot
Ok, well I tried to work with infinity as if it were a number, that was my first mistake.
My second was simplifying the function to say that the "...infinity" section would diverge because the sqrt(big# + big#)*big#=bigger#
But that's a mistake, because it's an infinite function and I can't really define a tail end for the series. When you replace the "...infinitiy" part with a big number, the value is always smaller than 2000. Which means that the series as a whole does converge.
Basically I didn't go about the problem the mathematical way. When I saw my mistake, I edited it out.