On series with alternating signs in the Euclidean metric (Q1852313)

Series of the form \[ \sum_{n=1}^\infty(-1)^{a_n} b_n\tag{1} \] are studied, where \((a_n)_1^\infty\in \{0,1\}^{\mathbb{N}}\), \(b_n\) are elements of a real Hilbert space. Mainly the structure of the sets \(C= \{(a_n)_1^\infty\in \{0,1\}^{\mathbb{N}}\): (1) converges\}, \(B= \{(a_n)_1^\infty\in \{0, 1\}^{\mathbb{N}}\): (1) is bounded\} are investigated in detail. Denote by \(M\) the class of all sequences \((a_n)_1^\infty\in \{0,1\}^{\mathbb{N}}\) after omitting sequences of the form \(a_1,\dots, a_n,0,1,1,1,\dots\). For \(a= (a_n)_1^\infty\in M\), put \(\phi(a)= \sum_{n=1}^\infty a_n 2^{-n}\); for \(a,b\in M\), let \(d_E(a, b)= |\phi(a)- \phi(b)|\). Then \((M,d_E)\) is a complete metric space. The author shows that the sets \(C\), \(B\) are sets of the first category in \((M,d_E)\) provided that there is a sequence \((a_n)_1^\infty\in M\) such that (1) diverges. If \(\sum_{n=1}^\infty|b_n|<+\infty\), then the Lebesgue measure of each of the sets \(\phi(C)\), \(\phi(B)\) is \(1\) and if \(\sum_{n=1}^\infty b_n= +\infty\), then this measure is \(0\).