Do we say a sum "diverges" or can we say it "equals $\infty$"?

Question

Is it correct or acceptable to say that a positive divergent series equals infinity or can we only say "it diverges"?

Ignoring the whole $-1/12$ thing where we assign finite values to divergent series, I'm not talking about that right now.

I'm asking about saying things like "$\sum_{n=0}^{\infty} n = \infty$" versus saying "The sum $\sum_{n=0}^{\infty} n$ diverges and that's all we can say about it, it doesn't sum to any particular value."

So I am asking about this idea that it doesn't sum to any finite value, i.e. doesn't equal anything, but then we say it equals infinity, which isn't a number.

I hope I'm asking this correctly. I'm mostly just interested if $\sum_{n=0}^{\infty} n = \infty$ is considered acceptable or if it's technically wrong terminology, and we can't assign anything to the sum and say it diverges.

Answer 1

Yes, you can say that a sum is infinite.

Proof: Gilbert Strang and Patrick Fitzpatrick say that in their books (see here and here).

Answer 2

It's equivalent to say "the series (or sequence) diverges" and $lim_n \sum_{i=1}^n a_i= \pm \infty$ (or $\lim_n a_n = \pm \infty$) or the limit doesn't exist at all.

Answer 3

Saying a sum "converges to $\infty$ (or $-\infty$)" is just saying that it diverges in a special way. But a sum may "diverge" without "converging to $\infty$ (or $-\infty$)- for example $\sum_{i=0}^\infty (-1)^i$ diverges but does not go to "infinity" or "negative infinity", it diverges because the sequence of partial sums, 1, 0, 1, 0, 1, 0,... does not converge.