Sum from infinity to infinity of infinity

Question

What is $\sum_{n=∞}^{∞}∞$? Indeterminate? Is the answer the same for $\sum_{n=∞}^{∞}k$ for any non-zero $k$?

This is probably a dumb and also useless question but I was just curious. Thanks!

Answer 1

Normally we require the summation variable to be an integer. $\infty$ is not an integer. We allow things like $\sum_{i=-\infty}^\infty$ as a shorthand for $\lim_{m \to -\infty} \lim_{n=\infty} \sum_{i=m}^n$ because it is useful. $\sum_{n=\infty}^\infty$ does not have a meaning that I recognize. If you define it clearly, the question will have an answer.