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Is it correct or acceptable to say that a positive divergent series equals infinity or can we only say "it diverges"?

Ignoring the whole $-1/12$ thing where we assign finite values to divergent series, I'm not talking about that right now.

I'm asking about saying things like "$\sum_{n=0}^{\infty} n = \infty$" versus saying "The sum $\sum_{n=0}^{\infty} n$ diverges and that's all we can say about it, it doesn't sum to any particular value."

So I am asking about this idea that it doesn't sum to any finite value, i.e. doesn't equal anything, but then we say it equals infinity, which isn't a number.

I hope I'm asking this correctly. I'm mostly just interested if $\sum_{n=0}^{\infty} n = \infty$ is considered acceptable or if it's technically wrong terminology, and we can't assign anything to the sum and say it diverges.

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  • $\begingroup$ this must not be, see this series: $$1-1+1-1+1-...$$ $\endgroup$ Feb 7, 2018 at 16:15
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    $\begingroup$ I would consider sequences first. Consider your definition of convergence of a sequence. That is one thing that can happen to a sequence. Consider now the negation of convergence. It turns out only two things can happen, the sequence either oscillates or it explodes ($\pm \infty$)... You might also be confused about what is the usefulness of assigning the value $\infty$ to an object, given that $\infty$ is not given many algebraic properties. $\endgroup$
    – Rgkpdx
    Feb 7, 2018 at 16:18
  • $\begingroup$ There are various definitions of limits, and the "infinite" case would only be one of them. But infinity is not a rational number, technically saying that the limit is "infinite" is wrong. Instead you would say that the series increases without bounds or something to that tune. Informally everyone would still understand what you are saying though. $\endgroup$
    – Tony
    Feb 7, 2018 at 16:23
  • $\begingroup$ I would say diverges or oscillates as $\infty$ is not a number, so that the = sign is not appropriate. $\endgroup$
    – Paul
    Feb 7, 2018 at 16:24
  • $\begingroup$ We can say "diverges" and then more specifically "diverges to $+\infty$" or "diverges to $-\infty$". It's a bit weird to say "converges to $+\infty$". A limit being equal to infinity is a perfectly well-defined concept in the extended reals. The only problem is that the extended reals have bad structure (all our basic arithmetic operations break down in certain situations). $\endgroup$
    – Ian
    Feb 7, 2018 at 16:24

3 Answers 3

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Saying a sum "converges to $\infty$ (or $-\infty$)" is just saying that it diverges in a special way. But a sum may "diverge" without "converging to $\infty$ (or $-\infty$)- for example $\sum_{i=0}^\infty (-1)^i$ diverges but does not go to "infinity" or "negative infinity", it diverges because the sequence of partial sums, 1, 0, 1, 0, 1, 0,... does not converge.

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  • $\begingroup$ I'm mostly asking about sequences/series (whatever the difference is) that increase forever, and then trying to ask what the sum of that is, if it's correct to say "it just diverges" or if we can say it "equals" infinity. $\endgroup$
    – user529129
    Feb 7, 2018 at 16:26
  • $\begingroup$ @user529129 A sequence is an indexed "vector" of elements. $(a_1, a_2, ..., a_n)$ is a finite sequence. A series is a sum over a sequence: So $\sum_{i=1}^n a_i$ is a series. $\endgroup$
    – mavavilj
    Feb 7, 2018 at 16:31
  • $\begingroup$ Is there such a thing as an infinite sequence? Or is it always finite by definition? $\endgroup$
    – user529129
    Feb 7, 2018 at 16:33
  • $\begingroup$ @user529129 Yes there is, just as there are infinite series, I was just using finite sequence as a simpler example. $\endgroup$
    – mavavilj
    Feb 7, 2018 at 16:36
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Yes, you can say that a sum is infinite.

Proof: Gilbert Strang and Patrick Fitzpatrick say that in their books (see here and here).

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It's equivalent to say "the series (or sequence) diverges" and $lim_n \sum_{i=1}^n a_i= \pm \infty$ (or $\lim_n a_n = \pm \infty$) or the limit doesn't exist at all.

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  • $\begingroup$ Is the idea of a limit "equalling" infinity an accepted definition / is it explicitly defined? Is the idea same for an infinite sum (i.e. not a limit)? $\endgroup$
    – user529129
    Feb 7, 2018 at 16:24
  • $\begingroup$ @user529129 It applies to all series and all sequences. Either the limit exists or it doesn't exist (in which case it doesn't exist or equals $\pm \infty$). $\endgroup$
    – mavavilj
    Feb 7, 2018 at 16:25

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