What is infinity divided by infinity?

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 $\frac{1+1+1+\ldots}{2+2+2+\ldots}=\frac12$?

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.

I will quote the following from Prime obsession by John Derbyshire, to answer your question.

Nonmathematical people sometimes ask me, “You know math, huh? Tell me something I’ve always wondered, What is infinity divided by infinity?” I can only reply, “The words you just uttered do not make sense. That was not a mathematical sentence. You spoke of ‘infinity’ as if it were a number. It’s not. You may as well ask, ‘What is truth divided by beauty?’ I have no clue. I only know how to divide numbers. ‘Infinity,’ ‘truth,’ ‘beauty’—those are not numbers.”