Taylor expansions and summation of series
The definition of \(\sum_{n\geq 0} a_n= \alpha_0\) in common sense, viz. Cauchy's, amounts to the existence of a null sequence \((\alpha_0, \alpha_1, \dots)\) such that \(\alpha_n- \alpha_{n+1}= a_n\), \(n\geq 0\). When termed \((\text{id--D}) \alpha= a\), this ``differential equation'' gives rise to an algebraic concept of summation, with \(a\), \(\alpha\) being in some appropriate \(\mathbb{C}\)-linear space where linear operators simulate differentiation (D) and summation (S), discrete or not, besides evaluation (V) that maps onto the most standard ``convergence elements''. Spaces thus furnished are called of type Taylor since their elements have Taylor-like ``algebraic expansions''. If, e.g., \(\sum_{n\geq 0} a_n\) has Abel limit \(\lambda\), that equation above with \(a: z\mapsto \sum_{n\geq 0} a_n z^n\) and \(\text{D} \sum_{n\geq 0} \alpha_n z^n:= \sum_{n\geq 0} \alpha_{n+1} z^n\) admits, within the space of holomorphic \(f(z)\), \(|z|<1\), for which \(f(1-0)\) exists, a unique solution \(\alpha: z\mapsto \sum_{n\geq 0} \alpha_n z^n\), rendering \(\lambda= \alpha (0)\). Here, at Abel's method, the Taylor space may be considered its convergence field, contrary to Cauchy convergence with a Taylor space of null sequences. In case of the Euler and the Borel means, D is a discrete resp. a proper differential operator, indicating the latter method to be a ``continuous'' counterpart of the former one. Similarly, through a shift D operating on a discrete resp. a continuous argument, convergence is related to a generalized limit as was introduced by Ramanujan (and makes Euler's constant sum the harmonic series). The new concept is also used to naturally conceive the notion of series products, notably the Cauchy product. The quotation from \textit{G. H. Hardy} is misleading, due to print setting on pp. 98, 99.
scientific article; zbMATH DE number 797133