This should be a simple question but I just want to make sure.
I know $\infty/\infty$ is undefined. However, if we have 2 equal infinities divided by each other, would it be 1?
And if we have an infinity divided by another half-as-big infinity, would we get 2? For example $\frac{1+1+1+\ldots}{2+2+2+\ldots}=\frac12$?
Best Answer
Essentially, you gave the answer yourself: "infinity over infinity" is not defined just because it should be the result of limiting processes of different nature. I.e., since such a definition would be given for the sake of completeness and coherence with the fact "the limiting ratio is the ratio of the limits", your
$$ \frac{1 + 1 + \cdots}{2 + 2 + \cdots} = \lim_{n \to \infty} \frac{n}{2n} = \frac{1}{2} $$
and, say (this is my choice)
$$ \frac{1 + 1 + 1 + \cdots}{1 + 2 + 3 + \cdots} = \lim_{n \to \infty} \frac{n}{n(n+1)/2} = 0 $$
would have to be equal (as they commonly define $\infty/\infty$), which does not happen.