You missed the lecture on the missuse of infinities.
If I have inf*k = inf,and dvide both sides by inf... ( The misuse) Then 1 = any K including 1/12. Now this is useless in calculus and number theory, but in quantium field theory it is a useful tool.
So inf = 1/12 and a non convergent series = a constant, but you have misused dividing infinity by itself to get it.
Infinity for division? It's useful, like counting chickens starting at zero. L'Hoptals rule is a very useful tool, but do not misuse it.
0^0 got Gemini 2.5 pro the other day for me. It claimed all indeterminate forms (in the context of limits) are also undefined as a response to a prompt dividing by zero. 0^0 is the most obvious exception, it's typically defined as =1 as you said.
0^0 = 1? Yes, it’s simpler that way.
0! = 1? Yes, it’s simpler that way.
0/0 = ∞? No, it’s undefined.
0.9999… = 1? Yes, it’s just two ways of expressing the same number.
1+2+3+… = -1/12? No, but if it did have a finite value, that’s what it would be.