Extending tests for convergence of number series
A nonnegative null sequence \(\{a_k\}\) is said to be weakly monotone, written WMS, if there exists a positive constant \(C\) such that \(a_k\leq Ca_n\) for any \(k\in[n, 2n]\). Similarly, a nonnegative function \(f\) defined on \((0,\infty)\) is said to be WMF if \(f(t)\leq Cf(x)\) for any \(t\in[x, 2x]\). In some problems, one should consider a smaller class than WM. A positive null sequence \(\{a_k\}\) is said to be generally monotone, written GMS, if there exists a constant \(C\) such that for any \(n\), \[ \sum^{2n}_{k=n} |\Delta a_k|\leq Ca_n,\quad\text{where }\Delta a_k:= a_k- a_{k+1}. \] Similarly, a nonnegative function is said to be GMF if \[ \int^{2x}_x |df(t)|\leq Cf(x)\quad\text{for any }x\in (0,\infty). \] Analyzing several classical tests for convergence/divergence of number series, the present authors relax the monotonicity assumption for the sequence of terms of the series. The following theorems are typical. Theorem 2.1 (Maclaurin-Cauchy test). If \(f\) is WMF, then the series \(\sum^\infty_{k=1}f(k)\) and the integral \(\int^\infty_1f(t)\,dt\) converge or diverge simultaneously. Theorem 3.2 (Cauchy condensation test). If \(\{a_k\}\) is WMS, then the series \(\sum^\infty_{k=1} a_k\) and \(\sum^\infty_{k=1} 2^k a_k\) converge or diverge simultaneously. Proposition 5.3. If \(\{a_k\}\) is GMS, then the series \(\sum^\infty_{k=1} a_k\) and \(\sum^\infty_{k=1} k|\Delta a_k|\) converge or diverge simultaneously. The sharpness of the results is also verified by presenting negative type results.
monotone/weakly monotone/generally monotone functions and sequences
scientific article; zbMATH DE number 5854869