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Articles 1 - 5 of 5
Full-Text Articles in Mathematics
The Basel Problem And Summing Rational Functions Over Integers, Pranjal Jain
The Basel Problem And Summing Rational Functions Over Integers, Pranjal Jain
Rose-Hulman Undergraduate Mathematics Journal
We provide a general method to evaluate convergent sums of the form ∑_{k∈Z} R(k) where R is a rational function with complex coefficients. The method is entirely elementary and does not require any calculus beyond some standard limits and convergence criteria. It is inspired by a geometric solution to the famous Basel Problem given by Wästlund (2010), so we begin by demonstrating the method on the Basel Problem to serve as a pilot application. We conclude by applying our ideas to prove Euler’s factorisation for sin x which he originally used to solve the Basel Problem.
A Construction That Produces Wallis-Type Formulas, Joshua M. Fitzhugh, David L. Farnsworth
A Construction That Produces Wallis-Type Formulas, Joshua M. Fitzhugh, David L. Farnsworth
Articles
Generalizations of the geometric construction that repeatedly attaches rectangles to a square, originally given by Myer- son, are presented. The initial square is replaced with a rectangle, and also the dimensionality of the construction is increased. By selecting values for the various parameters, such as the lengths of the sides of the original rectangle or rectangular box in dimensions more than two and their relationships to the size of the attached rectangles or rectangular boxes, some interesting formulas are found. Examples are Wallis-type infinite-product formulas for the areas of p-circles with p > 1.
Conditional Convergence Of Infinite Products, William F. Trench
Conditional Convergence Of Infinite Products, William F. Trench
William F. Trench
No abstract provided.
Invertibly Convergent Infinite Products Of Matrices, William F. Trench
Invertibly Convergent Infinite Products Of Matrices, William F. Trench
William F. Trench
No abstract provided.
Infinite Product Spaces Under The Tychonoff And Goofynoff Topologies, James A. Capps
Infinite Product Spaces Under The Tychonoff And Goofynoff Topologies, James A. Capps
All Graduate Plan B and other Reports, Spring 1920 to Spring 2023
The only topology considered for the infinite product of topological spaces in most current topology texts and research papers is the Tychonoff topology. Yet there is another topology which seems to be a much more topologically natural generalization of the usual "box" topology of finite products. We call this natural generalization the Goofynoff topology and exploit its properties. The use of the word "Goofynoff" (pronounced Goof'-n-off) is not universal and does not refer to any person of that name. In the few references to this topology that can be found, it is usually called simply the Box Topology. None of …