The irrationality of infinite series of a special kind (Q2836160)

For a sequence \(\{x_n\}_{n=0}^\infty \) of real numbers denote \(\nabla x_n=x_n-x_{n-1}\) and \(\nabla ^3 x_n=\nabla \bigl (\nabla (\nabla x_n)\bigr)\) for all integers \(n\geq 3\). The author proved the following theorems.NEWLINENEWLINETheorem 1: Let \(\varepsilon \in [0;1)\), \(\{a_n\}_{n=1}^\infty \) and \(\{b_n\}_{n=1}^\infty \) be sequences of positive integers such that NEWLINE\[NEWLINE\limsup_{n\to \infty}\left (\frac {a_{n+1}}{b_{n+1}}\frac 1{a_1a_2\dots a_n}\right)=\infty \;\text{and}\; \nabla ^3\bigl (\frac {n^\varepsilon a_n}{b_n}\bigr)\geq 0NEWLINE\]NEWLINE for every sufficiently large positive integers \(n\). Assume that \(\alpha =\sum_{n=1}^\infty \frac {a_n}{b_n}\) is convergent. Then the number \(\alpha \) is irrational.NEWLINENEWLINETheorem 2: Let \(\delta >0\), \(\varepsilon \in [0;1)\), \(\{a_n\}_{n=1}^\infty \) and \(\{b_n\}_{n=1}^\infty \) be sequences of positive integers such that NEWLINE\[NEWLINE\limsup_{n\to \infty}\left (\log \left (\frac {a_{n+1}}{b_{n+1}}\right)\frac 1{a_1a_2\dots a_n}\right)=\infty \;\text{and}\;\nabla ^3\Bigl (\frac {n^\varepsilon a_n}{b_n}\Bigr)\geq 0NEWLINE\]NEWLINE for every sufficiently large positive integers \(n\). Assume that \(\alpha =\sum_{n=1}^\infty \frac {a_n}{b_n}\) is convergent. Then the number \(\alpha \) is irrational and has irrationality measure greater than \(\delta \).