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God made the natural numbers; all the rest is
the work of man. - Leopold
Kronecker
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The purpose of Series Math Study (SMS) website is to explore and develop the beauty of series formulas that have connections to special mathematical constants. Most of the series in this website are found with completed proofs. Some series can be used to calculate ever more accurate values of some special math constants as the desire that has challenged mathematicians for many centuries.
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I am
ashamed to tell you to how many figures I carried these computations,
having no other business at the time. |
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Issac Newton, personal
journal, 1666
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(December 20, 2009) |
Infinite Series
in Connection with Pi Constants
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(November 26, 2009 – Happy Thanksgiving) |
Finite Series in Connection with Apéry, Pi Constants
The
n-th
partial
sum
of
the series
below
is
expressed
in
terms
of
Hurwitz
zeta function for each positive integer n.
,
where
,
s and a
are complex variables.
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(November 07, 2009) |
Finite Series in Closed Form
For real x and each positive integer n,
.
This finite series is also defined in the Riemann zeta function form. Read more >> .
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(October 25, 2009) |
Finite Series
in
General
Form
For real x ≠
0 and each positive integer n,
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(September 26, 2009) |
50 identities of Power Summation
(Update)
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(July 8, 2009) |
A Family of Finite BBP-Type Series in the Base of
729
For each positive integer n,
.
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(July 1, 2009) |
Some BBP-Type Series for Computing Pi (Update)
·
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·
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(June 14, 2009) |
A Brief Note of the Sum of Riemann Zeta Function
The sum of Riemann zeta function, 
,
is found
in the closed-form.
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(April 12, 2009 and May 03, 2009) |
For each positive integer n, the two finite series below are found in a closed-form.
.
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(April 9, 2009) |
Some finite
series help to find a family of
Machin-like
formula.
For each positive integer n,
.
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(December 14, 2008) |
An infinite series shows a connection between the quartic equation and the constant Pi.
,
where
and
it is
one of the roots of the quartic equation
.
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(December 13, 2008) |
The Riemann Zeta (7) constant has been found in a series in which the hyperbolic functions and other math constants appear.
,
where 
is
a
Riemann zeta constant.
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(August 12, 2008) |
The infinite series of the
BBP-type
formula is found and used to compute the constant 
.
.
Another series is found in terms of other math constants, namely
.
Click here to see other similar series of this type.
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(August 2, 2008) |
General Inverse Tangent Series of unknown names.
,
where 
, 
,
and 
.
It reveals many inverse tangent and Machine-like formulas. For example, the simple one of this type is obtained when n = 1, namely
.
Click here to see other forms.
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(May 1, 2008) |
Power Sum and Sum of Partial Power Sums for any positive integer n.
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Power Sum
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Sum of Partial Power Sums
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Click here to see other similar formulas.
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(Jan 31, 2008) |
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(May 15, 2007) |
The general series formula below
is true for all |x| 
a
and a 
1,
namely
.
A special case as a = 1 and x = 0, it gives
.
Click here to see other form.
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(April 07, 2007) |
The fast convergent series are used for computing the logarithm constants log 2 and log 3 (updated).
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(February 14, 2007) |
.
(Notice 
are
the
special values of the Riemann zeta function at positive integers.)
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(December 23, 2006) |
The formula below is true
for all
.
.
Click here to see more formulas and examples.
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(December 13, 2006) |
The reciprocal of the
beautiful infinite product of
nested radicals 
due
to Vieta
in 1592 can be decomposed into partial fractions of the infinite series
as
shown below or click
here.
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(October 15, 2006) |
Finite Alternative Odd Power Series
The sums of the following identities are true for any positive integer n.
1.
2.
3.
4.
More formulas and examples of this type are found in this link.
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