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Here, I am asking only about convergent series having partial sums $S_n$ which monotonically increase. In these cases, if I am not mistaken, all the partial sums fall in the closed interval $[S_1, S_\infty]$, but do they also fall in the half-open interval $[S_1, S_\infty)$?

One such series is$\sum\limits_{n=1}^{\infty} 9\left( \frac{1}{10} \right)^{n}$, which is used in this well-regarded reply justifying taking 0.999... as being equal to 1, which in turn suggests to me that the half-open interval $[0.9, 1)$ is insufficient in this case (at least if one accepts the caveats and conventions mentioned in the last paragraph of that reply.)

Update: I see I should have explicitly asked that, if the half-open interval is sufficient, then I would greatly appreciate some explanation for where I am mistaken in worrying about the series from the 0.999... = 1 answer (for example, if it has something to do with limits, would that also apply to the linked answer?)

sdenham
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    The half open interval is fine. $S_\infty > S_n$ for any $n$ that is finite (i.e., any $n$) – rubikscube09 Feb 03 '21 at 21:31
  • @rubikscube09 For all practical purposes, that seems fine, but when the topic is infinite series, I don't think we are just concerned with finite n, are we? – sdenham Feb 04 '21 at 15:40
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    all of the partial sums have $n$ finite. Things change when you take a limit, and that can be viewed as the transition from finite to infinite. – rubikscube09 Feb 04 '21 at 15:41
  • Regarding your notation $S_\infty$, I notice that you do not actually define what that notation represents. If your intention is that it represents the limit of the partial sums, which is equal to $1$, then that's fine. In that case $[S_1,S_\infty)$ and $[0.9,1)$ are two notations for the exact same interval, and every partial sum is contained in that interval. – Lee Mosher Feb 04 '21 at 16:03
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    Also, in your comment you seem concerned about "finite $n$", but keep in mind: $n$ represents a natural number, i.e. an element of the set ${1,2,3,...}$, and all natural numbers are finite. The symbol $\infty$ does not represent a number. So, we do not set $n=\infty$ and we do not refer to $S_\infty$ as a partial sum. Instead, $S_\infty$ represents the limit of the partial sums. – Lee Mosher Feb 04 '21 at 16:04
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    And finally, regarding your update, all rigorous interpretations of $0.999...=1$ have something to do with limits. The rest is just assuaging our wayward intuition. – Lee Mosher Feb 04 '21 at 16:06

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