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Inspired by The set of all limits of sub-series of an absolute convergent series is the following true?:

Let $a_n$ be a strictly decreasing sequence and $\sum_1^\infty a_n=\ell<\infty$ is a convergent series. Is it true to say that the set of all possible value of all subseries $\sum a_{n_i}$ of $\sum a_n$ is whole $[0,\ell]$?

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    $\begingroup$ If $a_n = 2\cdot(\frac{1}{3})^n$ (numbering from $n=1$) then the set in question is precisely the standard Cantor set (seen as the set of reals in $[0,1]$ having a base $3$ expression consisting only of $0$ and $2$), so, no. $\endgroup$
    – Gro-Tsen
    Commented Mar 23, 2023 at 22:17
  • $\begingroup$ @Gro-Tsen Thank you for this example $\endgroup$ Commented Mar 23, 2023 at 22:19
  • $\begingroup$ @Gro-Tsen so under what condition the answer is affirmative? $\endgroup$ Commented Mar 23, 2023 at 22:22
  • $\begingroup$ @Gro-Tsen Is $\sum (1/2^n) $ is some what an exception? I mean what can be said about the space of all positive sequence whith connected subseries values? $\endgroup$ Commented Mar 23, 2023 at 22:38
  • $\begingroup$ @Gro-Tsen your interesting example of the Cantor set is a motivation to assigne a real number to every positive terms series: the Hausdorff dimension of the set of all subseries values. $\endgroup$ Commented Mar 24, 2023 at 1:46

2 Answers 2

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For convenience define $S_n = \sum_{j\le n} a_j$ and $T_n = \sum_{j > n} a_j$.

Suppose there is $n$ such that $a_n > T_n$. Then for $S_{n-1} + T_n < x < S_n$, $x$ is not the sum of a subseries.

On the other hand, if $a_n \le T_n$ for all $n$, then every $x \in [0,\ell]$ is the sum of a subseries. This can be obtained "greedily": include $a_n$ iff the sum of $a_n$ and already-included terms $\le x$.

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  • $\begingroup$ Thank you very much for this perfect answer $\endgroup$ Commented Mar 26, 2023 at 11:22
  • $\begingroup$ Is there a terminology for such kind of sequences? may be a reference for such kind of seuqnces? $\endgroup$ Commented Mar 27, 2023 at 12:19
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No. Consider the sequence $1 + 10^{-10} + 10^{-100}+10^{-1000}+ \cdots$. You can't possibly find a subsequence adding up to $1/2$, because the tail is so small.

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  • $\begingroup$ Thank you for your answer. Under what condition the answer is affirmative? is there a terminology for positive series with connected set of all possible sub series value? $\endgroup$ Commented Mar 23, 2023 at 22:32
  • $\begingroup$ Is $\sum (1/2^n) $ is some what an exception? I mean what can be said about the space of all positive sequence whith connected subseries values? $\endgroup$ Commented Mar 23, 2023 at 22:37

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