Given a real valued function, on a closed interval [a,b].

Define the sum of all real numbers f(x) from a<x<b, inclusive. I gave it a definition which looks precisely like the Riemann Integral with the delta-x part missing. Thus it is the sum f(a+(b-a)/n*k) from k=1 to k=n (just like the Riemann sum with the delta-x part missing) with the limit as n tends to infinity. The question I have is does this type of "sum" ever converge?