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Related SMS

Dear Readers,

 
Series Math Study (SMS), founded in 2005 by Tue N. Vu, is a non-profit and a personal Web site.  The purpose of SMS is to explore and develop the beauty of series formulas that have connections to special mathematical constants such as Archimedes, Natural Logarithmic Base, Euler, Catalan, etc. Some series contain special structures to reveal themselves of fast convergent series that can be used to calculate ever more accurate values of those special mathematical constants to billion places. In this website SMS does not use programmatic algorithms and powerful mathematical computing environments like Mapple and Matlab to research.   SMS pursues using heuristic mathematics to develop the general structures of series.
 
Math series are always the beautiful forms in mathematics in which the nature numbers play an important role, especially for those numbers hidden under the convergent series can provide quite useful analytical approximations. Since series are used in many branches of mathematics, and understanding of convergence of series is necessary. It is useful to classify the various types of fast convergent series and constitute some methods to determine an exact value of each sum or product series in terms of special mathematical constants in a closed form. The mathematical tools such as calculus, number theory and others can be used to develop each general structure and series formula. When a general series formula is determined in a closed form, its exact value is evaluated in terms of other mathematical constants. A series that its exact value is determined is used as the fundamental units to build other modular series. Some series, which its exact values are not determined, can potentially become math constants if its general structures are not existed in any mathematical forms.  The mysterious series are always the topics of interest in mathematics to encourage the curiousness of the next generations. The desire of understanding how those mysterious series are constructed might lead discovery of new series.  That tend is a substance to help developing systematic method such that it can determine exact values of series in closed-forms and provide a way to evaluate approximate values of certain important series. The method of evaluating of a convergent series to give an approximate value is considered as one of the initial steps in gradually opening a secret structure of that series in terms of math notations and functions, which can be used to establish its general structure and model. Some interested convergent series are deeply studied to indicate whether its values are irrational or transcendental or whether it has connections to the special math constants Pi, e or other known ones. The searching for the closed-forms of the interested series is always a dynamic force to help exploring new approaches, which are built on top of the current fundamental mathematics, to establish special mathematical models in future. It is known that the math constants Pi and e are irrational and transcendental numbers through centuries because they are determined based on geometry, Cartesian, Polar or Parametric coordinate system. In a certain aspect there is a question whether all math constants can be expressed in normal integer numbers if a new transcendental mathematical system is developed while the nature or integer numbers are considered to be irrational? The reason for asking is merely to show a truth that numbers and mathematical models is an intelligent game of the Creator.
 
Even if the mathematicians have known many series, that knowledge would be still a drop in ocean compared to the innumerable series that had been hidden under multi-forms and multi-structures in mathematics. The secrets of unknown series are gradually revealed in each era by challenging the works of men. In this Website SMS voluntarily sends to readers as a math gift, and if possible, to mathematics as a partial contribution even if it is a lone grain of sand, respectively. 
 
The most series found under “Series Outlines” section in this Web site were proved step by step. However, you are encouraged to verify the liability of specific formula if interested. The classification of series formulas to similar families or to related groups has not been considered because more coming series are often going to be “randomly” added and posted on the site without following specific rules.
 
 
Sincerely,
 
SMS
 

  

The trace of footsteps is marked on an expanse of sand, the wind can make it back to where it is. (T.V.)