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,...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.展开更多
Understanding microbial growth is essential to any research conducted in the fields of microbiology and biotechnology. Current methods of determining growth characteristics of microbes involve subjective graphical int...Understanding microbial growth is essential to any research conducted in the fields of microbiology and biotechnology. Current methods of determining growth characteristics of microbes involve subjective graphical interpretations of linearized logarithmic data. Reducing error in logistical data decreases disparity between graphical and analytical predictions of microbial characteristics. In this study, a method has been developed to calculate the kinetics of microbial characteristics utilizing a modified Maclaurin series. Convergence of this series approaches the true kinetic value of microbial characteristics to include specific growth rates. In this research, a modified Maclaurin series is used to evaluate microbial kinetics in comparison to graphical determinations.展开更多
文摘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.
文摘Understanding microbial growth is essential to any research conducted in the fields of microbiology and biotechnology. Current methods of determining growth characteristics of microbes involve subjective graphical interpretations of linearized logarithmic data. Reducing error in logistical data decreases disparity between graphical and analytical predictions of microbial characteristics. In this study, a method has been developed to calculate the kinetics of microbial characteristics utilizing a modified Maclaurin series. Convergence of this series approaches the true kinetic value of microbial characteristics to include specific growth rates. In this research, a modified Maclaurin series is used to evaluate microbial kinetics in comparison to graphical determinations.
