Characterization of first-order convergent sequences (Q1347169)

A sequence \((s_ n)\) of complex numbers with limit \(s\) is first-order convergent if \[ \lim_{n \to \infty} \left( {s_{n+1} - s \over s_ n - s} \right) = \sigma, \] with \(0 < | \sigma| \leq 1\), and if \(\sigma = 1\) then \[ \lim_{n \to \infty} \left( {s_{n+1} - s_ n \over s_ n - s_{n-1}} \right) = 1 \] holds as well. The convergence is logarithmic if \(\sigma = 1\), linear if \(0 < | \sigma | < 1\), higher-order if \(\sigma = 0\), and is called boundary linear in this paper if \(| \sigma | = 1\) but \(\sigma \neq 1\). The terms of the sequence may be interpreted as partial sums of a series \(s_ n = a_ 0 + a_ 1 + \cdots + a_ n\), and the paper characterizes first-order convergent sequences \((s_ n)\) by obtaining a complete description of the relation between \[ \lim_{n \to \infty} {s_{n+1} - s \over s_ n - s} \quad \text{and} \quad \lim_{n \to \infty} {a_{n+1} \over a_ n}. \] For the case of linear convergence these two limits were known to be equal. Here the boundary linear and logarithmic cases are examined. In addition to obtaining full characterizations, the author gives examples to show that these characterizations are optimal in the sense that the hypotheses of the lemmas involved cannot be weakened. The author then goes on to define the normalized generating function \(\sum_ n (a_ n/ \sigma^ n) z^ n\) for the sequence \((s_ n)\), and relate its properties to first-order convergence of \((s_ n)\).