Is there a shorthand notation for stating a series is convergent/divergent?

Question

Is there any way to condense a statement something like:

"If $\sum a_n$ is divergent, then $\sum |a_n|$ is divergent."

into a smaller amount of space?

Specifically, I am asking if there exists shorthand notation for quickly writing "is divergent" and "is convergent."

Answer 1

Not sure if this is technically "condensed" but if you want to use unambiguous notation:

Let $\Bbb{D}$ be the set of all divergent series.

$$\sum a_n\in\Bbb{D}\implies\sum |a_n|\in\Bbb{D}$$

This may come in handy especially if you will be using $\Bbb{D}$ throughout your paper or proof.

Hope this helps!

Answer 2

I'm thinking something similar to what Mad Max said in his response would work. Maybe it would be notationally simpler to say $\sum a_n \to c$ for convergent series, where c is just an arbitrary finite constant. For divergent sums, we could maybe say $\sum a_n \not\to c$. These aren't set in stone, but I'd say they have the potential to be useful notationally.

For example, your statement in this new notation would look like such:

If $\sum a_n \not\to c,$ then $\sum |a_n| \not\to c,$ for c $\in$ $\mathbb{R},\mathbb{C},$etc.