Is the sum of all rational numbers between two integers infinity If there are infinite numbers between two rational numbers then would that entail that the sum of all numbers, say between 1 and 2, be infinity?
I believe that this cannot be true and has to do something with area under a curve?
 A: Before you can answer this question you need a definition for what it means to "add up infinitely many numbers".
For the moment, assume those numbers are listed in some order:
$$
a_1, a_2, \ldots .
$$
Then mathematicians define the infinite sum to be the limit (if it exists) of the numbers
$$
a_1,  \quad  a_1 +  a_2, \quad  a_1 +  a_2 + a_3,  \ldots .
$$
Then, for example, you could show that the "infinite sum"
$$
1 +
\frac{1}{2} +
\frac{1}{4} +
\frac{1}{8} + \cdots
$$
is $2$. So sometimes the sum of infinitely many numbers is finite.
With that definition, you clearly can't sum all the rational numbers between $1$ and $2$ since all of them are greater than $1$, so those partial sums will grow without bound. You can't sum all the rational numbers between $0$ and $0.0001$ since infinitely many of them are greater than $0.00001$. So you are more or less correct - mathematicians prefer to say you can't sum them, not that the sum is infinity.
When you learn calculus (that "something with area under a curve") you will see how to add up more and more and more pieces without growing to infinity because at each stage the pieces are smaller and smaller and smaller.
