On series with alternating signs in the Euclidean metric.(English)Zbl 1011.40002

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 $$0$$.

MSC:

 40A05 Convergence and divergence of series and sequences