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I am writing a paper in precalculus about series, and one of the sentences I read from arithmetic series is this:

When dealing with arithmetic series, we take only the partial sum. This means that we will be considering only finite series since the terms in an arithmetic series is being obtained by adding the common difference. Thus, the next terms for an infinite arithmetic series will become infinitely large or infinitely small, and its sum cannot be determined.

I get that the sum will become infinitely large if $d > 0$. An example would be $a_1 = 1, d = 1$. What I don't get is that the sum will become infinitely small. It seems like the author is referring to a sum that decreases without bound like $a_1 = 1, d = -1$.

I also think that the term is used in limits. For instance, if $|x - 1| < \varepsilon$ and $\varepsilon$ approaches zero, then $|x - 1|$ be infinitely small and will also approach zero.

However, we also tend to say that something is small by subtracting. Let's say 10 is a large number. To make it smaller, we can subtract 5 to make it smaller, which is now 5.

Is infinitely small referring to a value that is much less than zero?

Edit: I am asking about the usage of 'infinitely small' whether it should refer to the value closer to zero, or values decreasing without bound.

soupless
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    More context would be helpful. “Infinitely small“ can mean “arbitrarily close to zero” or “arbitrarily close to $-\infty$”. – Martin R Nov 21 '21 at 09:04
  • @MartinR Is the sentence I posted not enough? If so, I'll try to add more. – soupless Nov 21 '21 at 09:05
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    Does this answer your question? How can I define an infinitely small positive value? – Dietrich Burde Nov 21 '21 at 09:06
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    I'm pretty sure that here, "becomes infinitely large" means approaches $+\infty$ and "becomes infinitely small" means approaches $-\infty.$ Incidentally, for sequences in general (but not for arithmetic or geometric sequences), note that there might be difference between the vaguely worded expressions "becomes infinitely large" and "becomes and remains infinitely large". – Dave L. Renfro Nov 21 '21 at 09:07
  • Oops, in my previous comment "but not for arithmetic or geometric sequences" should be "but not for arithmetic sequences", since the distinction I allude to DOES arise with geometric sequences, for instance when the common ratio $r$ satisfies $r < -1.$ – Dave L. Renfro Nov 21 '21 at 09:18
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    Actually the statement “the next terms for an infinite arithmetic series will become infinitely large or infinitely small” makes no sense to me. – Martin R Nov 21 '21 at 09:48
  • Adding more terms to the partial sum is probably what it means. – soupless Nov 21 '21 at 09:52

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It is an unfortunate ambiguity that "small" means both "close to zero" and "close to $-\infty$". They are synonymous in the context of positive numbers, but when negative numbers are allowed the two don't coincide.

Since we're talking about arithmetic sequences and series, (and, as I gather, trying to make sense of adding sensible, "natural" terms after the infinite sum) they can't keep getting closer and closer to zero indefinitely. So I guess they mean "close to $-\infty$".

Arthur
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