You can always break up ∞/∞ into the left hand side, where n is an arbitrary number. However, on the left hand side n∞ is always equal to 0. Thus ∞∞ should always equal 0.
How do you know 0+0+0+0+......=0? If you think about it 0+0+0+.....=0×∞=0×1/0=1. (or any other number). We clearly are dealing with a value on which standard arithmetic doesn’t apply. (Hence “infinity is not a number”.)
So the question becomes what does apply and how do we deal with this? And that is not an easy/simple question. It’s not hard… but it’s not simple. Bottom line, finite rules of arithmetic do not always apply, and such instincts lead to common traps.