The phi-ratio tests (Q1941567)

As concerns \(\sum 1/n^p\), the ratio test of d'Alembert and Cauchy is inconclusive if and only if \(p > 0\) (cf. the wording at ``Example 1''!); except for \(p = 1\), the test under consideration proves conclusive. It rests upon \(\varphi : \mathbb N \to \mathbb N := \{1, 2, \dots\}\) such that \(n/\varphi(n) \to \alpha \in (0,1)\) and applies to series with decreasing terms \(a_n > 0\). The ``\(\varphi\)-ratio test'' says: Whenever \(a_{\varphi(n)} /a_n \to L\) then \(\sum a_n\) converges if \(L < \alpha\) and it diverges if \(L > \alpha\); in case of \(L = \alpha\) the test is inconclusive. A precursor of this article (see there for \(m > 1\) in the present one) is [\textit{S. A. Ali}, Am. Math. Mon. 115, No. 6, 514--524 (2008; Zbl 1168.40002)].