On the algebraic equivalence of Taylor-Maclaurin series and the J. Bernoulli “universal series”
At first the issue of the equivalence between the Taylor series and the J. Bernoulli «series universalissima» was discussed in 1716 after the well-known publication of Taylor in 1715, and their equivalence was not in doubt even then. A detailed study of I. Bernoulli series N.Ya. Sonin published in 1897. An important part of his work is the issue formulation of the origins of the Taylor formula. N.Ya. Sonin believed that it was J. Bernoulli’s series, and Bertrand was of an opinion that it was Newton’s binomial. The paper proposes to search for the answer to that question in the algebra A  F ( P  s ), the algebra of all linear mappings of the space of polynomials in S formal generators over an arbitrary field F of characteristic zero into itself. Elements of that algebra are represented by non-commutative series in powers of differentiation operators with polynomial coefficients of the operators of multiplication by a variable over the field F . Then the series of J. Bernoulli, Taylor, Maclaurin and others are equivalent and can be interpreted as algebraic equalities in A  F ( P  s ).