A remark on the Abel's and Dirichlet's criterions concerning generalizations to monotonicity
The authors study particular generalizations of monotonicity for numerical sequences: quasi-monotonicity and rest bounded variation. A non-negative sequence \(A=\{a_n\}_{n=0}^{+\infty}\) is called quasi-monotone (\(A \in CQMS\)) if \(\exists \alpha \geq 0 \) such that the sequence \(\{\frac{a_n}{n^{\alpha}}\}_{n=1}^{+\infty}\) is decreasing; a non-negative null sequence \(A=\{a_n\}_{n=0}^{+\infty}\) is called of rest bounded variation (\(A \in RBVS\)) if \( \exists M(A) \) such that \( \sum_{n=m}^{+\infty}| a_n - a_{n+1}| \leq M(A) a_m\,\). (These two properties are not comparable as it was shown by \textit{L. Leindler} [Anal. Math. 28, No. 4, 279--286 (2002; Zbl 1026.42007)]). The authors prove the following theorems: (1) (generalization of Abel's and Dirichlet's classical criteria) if \(A=\{a_n\} \in RBVS\) and \(\{\sum_{k=1}^n b_k\}\) is bounded, then \(\sum_{n=1}^{+\infty} a_n b_n \) converges; (2) (generalization of Abel's and Dirichlet's classical criteria for uniform convergence) if \(A=\{a_n(x)\} \in RBVS \;\forall x\in D\) is uniformly bounded in \(D\) and \(\sum_{n=1}^{+\infty} b_n(x)\) converges uniformly in \(D\), then \(\sum_{n=1}^{+\infty} a_n(x) b_n(x)\) converges uniformly in \(D\); (3) (generalization of Abel's and Dirichlet's classical criteria for uniform convergence) if \(A=\{a_n(x)\} \in RBVS \;\forall x\in D\) tends to zero uniformly in \(D\) and \(\{\sum_{k=1}^n b_k(x)\}\) is uniformly bounded in \(D\), then \(\sum_{n=1}^{+\infty} a_n(x) b_n(x)\) converges uniformly in \(D\,\). Moreover, the authors give examples of (i) a bounded sequence \(A\in CQMS\) and a sequence \(\{b_n\}\) such that \(\sum_n b_n \) converges and \(\sum_n a_nb_n\) diverges; (ii) a bounded sequence \(A\in CQMS\) with \(\lim_n a_n = 0\) and a sequence \(\{b_n\}\) such that \(\{\sum_{k=1}^n b_k\} \) is bounded and \(\sum_n a_nb_n \) diverges. Similar examples are given for uniform convergence. In conclusion, Abel's and Dirichlet's criteria cannot be proved replacing monotonicity with quasi-monotonicity but only with rest bounded variation.
scientific article; zbMATH DE number 5857788