Dear Readers,
Series Math Study (SMS), a group of students,
has built nonprofit Web site in the year of 2005 with the purpose of exploring
the beauty of math series by using the knowledge of calculus and number theory techniques
to find new series and verify some series formulae that had been discovered
from previous centuries to present.
There are many math series that have connections to special constants
such as Archimedes, Natural Logarithmic Base, Euler, Catalan, etc. Some formulae 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
without computing any prior digits. 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.
We are interested not only in
exploring some fast convergent series but also in constituting some methods to
determine the exact value of each sum or product series in terms of special
mathematical constants. Once the exact
value of the series is determined it is called a closed form. The hardest thing when developing a series is
to find its exact value. As mentioned
above, we use calculus and number theory to develop series formula by
establishing its general structure and then solve for its general formula. When its general formula is determined, the value
of that specific series can be evaluated exactly in terms of other mathematical
constants. Whenever a series is
expressed in closed form such as even zetas series ( 
,
,
,
…), that series would be beautiful and valuable in math. The value of zeta (2)
was first discovered by Swiss mathematical genius Leonahard Euler in 1734. There are many math series with their exact
values are not known such as odd zetas ( 
,
,
,
…) but its approximated values can be calculated. The approximated values of those series can
be assigned to become new mathematical constants if its structures do not exist
in any mathematical forms. It is
interested to consider whether the approximated values of those series are
known to be irrational or transcendental.
Some special constants such as pi or e are real numbers, and they are
both known to be irrational and transcendental through centuries. We can say that the constant pi or e is
generated from shapes or numbers when considering it in either geometry or math
series under Cartesian coordinate system.
There is a question whether these math constants can be expressed in
normal integer numbers if a new math system is developed while the nature or integer
numbers are considered to be irrational?
When studying about series, we can understand how the nature numbers
hold an important role in math, and man creates formulae or other works. We understand that convergent series provide
quite useful analytical approximations.
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 curiosity of
understanding what is unknown is a main key to lead the discovery of many
interesting results. 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
knowledge of the unknown series would be indicated the time span of each
era. In this Web site, some series may
have not been found in math handbooks or from elsewhere. 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 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.