Generalized Series of Bernoulli Type  

作　　者：Thomas Beatty Nicholas Bianco Nicole Legge Thomas Beatty;Nicholas Bianco;Nicole Legge(Department of Mathematics, Florida Gulf Coast University, Fort Myers, FL, USA)

机构地区：[1]Department of Mathematics, Florida Gulf Coast University, Fort Myers, FL, USA

出　　处：《Advances in Pure Mathematics》2023年第9期537-542,共6页理论数学进展（英文）

摘　　要：The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.

关 键 词：BERNOULLI SERIES CONVERGENCE SUM Recursion Formula ZETA-FUNCTION SINE Maclaurin Series Infinite Product 

分 类 号：O17[理学—数学]