What is the negation of a converging series? 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?
 A: 
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?

Your safest bet (to be sure you cover all the bases, and be understood by all):
Take the contrapositve of the corollary to be "If $\lim_{n\to \infty}a_n \ne 0$, then the series $\sum_{n=m}^\infty a_n$ DOES NOT CONVERGE."
Sums that diverge still exist. They simply do not converge to any particular (finite) value.

Just an added note: In "Baby Rudin" (Principles of Mathematical Analysis), a diverging sequence is defined as a sequence which does not converge. A divergent series is a series for which the terms being summed form a diverging sequence.
A: Let $$s_k := \sum_{n=m}^k a_n$$ the partial sum of the series.
By definition the series $\sum_{n=m}^\infty a_n$ converges if and only if the limit of the partial sums $\lim_{k \to \infty} s_k$ exists and is finite.
So if $a_n$ doesn't converge to 0, the limit $\lim_{k \to \infty} s_k$ does not exist or is infinite. This is in my oppinion the best way to phrase the contrapositive.
