Thread: The Sum of all real numbers

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?

i belive it will never converge unless the intire function is zero (exept a finite number of values)
or
that the function is un-even in weak form ( f(c-x)=-f(c+x) for x in[a,b] or (a,b)* when c = (a+b) / 2 the sum is f(c) )

*if the interval is closed only in one side then add this side to the sum

I completely agree with you. Thus, we cannot define such a concept as the "real sum".