Dear Readers,
Series Math Study (SMS), founded in 2005 by
Tue N. Vu, is a non-profit 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
some special mathematical constants to billion places. 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 “Index” 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.
Comments and constructive suggestions are
always welcome.
Sincerely,
SMS
The trace of
footsteps is marked on an expanse of sand, the wind can make it back to where
it is.