I'm reading chapter 14 of Elementary Analysis by Ross.
A series that does not converge is said to diverge. We say that $\sum_{n=m}^na_n$ diverges to $+\infty$ and we write $\sum_{n=m}^\infty a_n=+\infty$ provided that $\lim s_n=+\infty$; a similar remark applies to $-\infty$. The symbol $\sum_{n=m}^\infty a_n$ has no meaning unless the series converges or diverges to $+\infty$ or $-\infty$.
The first sentence says the negation of a converging series is a diverging series. But I would say that this is an diverging series or a series that doesn't exist. For example if I want to take the contrapositive of this Corollary:
14.5 Corllary If a series $\sum_{n=m}^\infty a_n$ converges, then $\lim a_n = 0$
I would say that the contrapositive would be:
If $a_n$ doesn't converge to $0$, then the series $\sum_{n=m}^\infty a_n$ diverges or doesn't exist. Is this correct ? Or should I leave the part "or doesn't exist" out of the statement?