This was asked earlier at MSE.

Let $A = \{a_0, a_1, a_2, \dotsc\}$ denote a weakly decreasing sequence of positive terms whose sum converges. Next introduce plus minus signs in every possible way, to form the collection $L(A) = \{\sum \limits_{j=0}^\infty \epsilon_j a_j : \epsilon_j = \pm1\}$ of values of the signed sums. All of these series converge since they converge absolutely.

For example, let $a_j = 1/j!$, giving the usual series for $e = 2.718\ldots$. Then one can check that $L(A)$ has the following general properties: it is a proper closed uncountable subset of the interval $[-e,+e]$.

Questions: (1) Is it possible to be more specific and to say exactly what the set $L(A)$ actually is (apart from just recapitulating the definition)?

(2) Can one recover the sequence from the values of its signed sums? Or does there exist some $B = \{b_0, b_1, b_2, \dotsc\}$ satisfying (as above) $b_j \geq b_{j+1} > 0$ and such that $L(B) = L(A)$?

Thanks

`a$_j$`

(as opposed to`$a_j$`

). I have edited accordingly. If you really want upright letters, then I believe that the accepted convention is to use`\mathrm`

, like`$\mathrm a_j$`

($\mathrm a_j$). $\endgroup$