Q:

What is the limit of the ratio of the sum of all real numbers from 0 to 2 over the sum of all real numbers from 0 to 1. 

I approached this question I made, by saying that the sum of a finite amount of numbers from a to b separated by a common change is average•(# of numbers we have).  I found the formula the # of the numbers we have to be $\frac {b-a+n}{n}$, were n is the common change from one number to the other.  I also found the formula of the average to be $\frac {a+b}{2}$. Using these formulas I set up the formula of the ratio and  took the limit as n approaches 0 I got the ratio to be exactly 4.  Do I have justification for this, or is this complete nonsense.  Why or why not. 

A:

I would tentatively agree that the solution would be 4.  The $\int_0^2 x\ dx$ (which would be the sum of the reals) would be $x^2$ evaluated from 2 to 0 $(2^2 - 0^2) = 4$. Likewise, the $\int_0^1 x\ dx = 1^2 = 1$. Therefore the ratio would be $\frac{4}{1}$

