# 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?

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What on earth does it mean for a series to not exist? – Chris Eagle Dec 29 '12 at 16:15
Yes, that is the contrapositive, given your definitions. – Calvin Lin Dec 29 '12 at 16:16
I would say that if this limit doesn't exist: $\lim_{n\to\infty}\sum_{k=m}^n a_k$ My book means with a limit that doesn't exist a limit that doesn't take the value $+\infty$,$-\infty$ or a real number. – Kasper Dec 29 '12 at 16:19
You seem to be using "diverges" as a synonym for "diverges to $\infty$ or $-\infty$. The standard meaning of "diverges" is "does not converge to a real number." But there are some differences in usage. – André Nicolas Dec 29 '12 at 16:20
@Kasper: It is incorrect to say "the converse of a converging series is a diverging series". It is a definition; there is nothing to take a converse of. You mean to say "opposite" or "negation" perhaps. However, it is correct to talk about the converse to your stated corollary about convergent series. Please edit your question to make this clearer. – Zev Chonoles Dec 29 '12 at 16:43

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.

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Note that his definition of diverges is tends to $\pm \infty$. It doesn't account for the case where the sum bounces around, like in $a_n = (-1)^n$ which has partial sums -1, 0. In this case, he calls it that the limit doesn't exist. – Calvin Lin Dec 29 '12 at 16:21
He chose a very weird definition: "The symbol $\sum_{n=m}^\infty a_n$ has no meaning unless the series converges or diverges to $+\infty$ or $-\infty$". I can't say that I've seen it defined this way before. As such, the case where it bounces around is doesn't exist / not defined, by definition. – Calvin Lin Dec 29 '12 at 16:24
Im only citing my book. But I guess this definition is stated a little bit weird. – Kasper Dec 29 '12 at 16:29
No problem, Kasper. I think your question is legitimate! – amWhy Dec 29 '12 at 16:30
I'm still a little bit confused by this part: "The symbol $\sum_{n=m}^\infty a_n$ has no meaning unless the series converges or diverges to $+\infty$ or $-\infty$." What is meant with: "has no meaning" ? – Kasper Dec 29 '12 at 16:44

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.

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