# The set of all possible values of subseries of a convergent positive term series

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 $$[0,\ell]$$?

• 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. Commented Mar 23, 2023 at 22:17
• @Gro-Tsen Thank you for this example Commented Mar 23, 2023 at 22:19
• @Gro-Tsen so under what condition the answer is affirmative? Commented Mar 23, 2023 at 22:22
• @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? Commented Mar 23, 2023 at 22:38
• @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. Commented Mar 24, 2023 at 1:46

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 $$\le x$$.

• Thank you very much for this perfect answer Commented Mar 26, 2023 at 11:22
• Is there a terminology for such kind of sequences? may be a reference for such kind of seuqnces? Commented Mar 27, 2023 at 12:19

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.

• 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? Commented Mar 23, 2023 at 22:32
• 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? Commented Mar 23, 2023 at 22:37