Series Summary provides the outlines and concise notes regarding the series and formulas related to the posts that have been published on the Series Math Study website.
Special Finite Series
Special Arctan Identity
where x is arbitrary.
Pi/35
The sin function of pi/35 obeys the identity
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Special Sextic Equation
where a, b and c are real, and a ≠ 0. One of the roots of the special sextic equation is found as shown below
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The solution of the equation is
December 30, 2012 |
A Special Series Involving Gamma Function
We found a new special series formula in connect with Gamma function Γ(x). For real x,
The formula can be rewritten in the form
where x < -1/2 or x > 1/2.
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August 02, 2012 |
A Special Series
where
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July 29, 2012 |
Some Special Series
where
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June 2, 2012 |
A Closed Form of Special Value of Gamma Function
where m is a positive integer.
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May 20, 2012 |
A Special Limit Expression Involving Gamma Function
where x is real.
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January 05, 2012 |
New Formula of Gamma function Approximation The formula below provides an approximation of the Gamma function, offering a high degree of accuracy for real values of
Notes 1. The left-hand side represents the natural logarithm of the Gamma function. 2. In mathematics, the Gamma function is crucial as many other special functions depend on it. Discovering a closed-form solution for the Gamma function would significantly enhance the chances of resolving the Riemann Hypothesis. Unfortunately, such a closed form remains unknown. |
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November 06, 2011 |
A New Approximation Formula for Computing the N-th Harmonic Number A newly derived approximation formula, which provides more digit accuracy for computing the n-th Harmonic Number, is found as follows:
where
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June 24, 2011 |
New Harmonic Number Approximation Formula
where
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January 02, 2011 |
A special product series gives
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where Γ is Gamma function, and Γ(1/3) = 2.6789385347... .
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October 19, 2010 | ||||
Gamma Function Approximation Formula (6 decimal places) The following formula is the Gamma function approximation that provides a high level of accuracy. It gives the value of Gamma function to 6 decimal places of precision for real x < 10, namely
(Notice, the exponential function is written as exp(x) or ex. This formula is also expressed in terms of logarithm to compute complex z.)
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October 16, 2010 | ||||
Two Series are found in Connection with Mathematical constants,
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August 03, 2010 | ||||
Two Series in Connection with Mathematical constants
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July 03, 2010 | ||||
Sums in which the Square Root of Two and Other Constants Appear are Given by
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June 27, 2010 | ||||
A Surprising Sum in which![]() |
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April 18, 2010 | ||||
Some Special Infinite Series | ||||
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February 09, 2010 | ||||
Sums of Reciprocals of Two-Term Squares Table | ||||
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Sums of Reciprocals of Two-Term Cubes Table
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January 30, 2010 | ||||
Series in Limit Form |
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Given n is a positive integer. The following series are found in a closed form as n tends to infinity.
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January 17, 2010 | ||||
Sequences and Series Art -- A Generic Infinite Series Found Linking Three Special Sequences '2, 30, 420, ...', '15, 209, 2911, ...', and '17, 241, 3361, ...' with the Constant
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January 08, 2010 | ||||
Two Beautiful Series in Connection with zeta ![]()
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January 01, 2010 | ||||
A Curious Series in Connection with Euler-Mascheroni, Pi Constants For positive integer n,
The above series can be transformed into another form, namely
Another curious series is found in connection to Pi
(Happy New Year 2010) |
December 20, 2009 |
Infinite Series in Connection with Pi Constant
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November 26, 2009 (Happy Thanksgiving) |
Finite Series in Connection with Apéry, Pi Constants
The n-th partial sum below is expressed in terms of Hurwitz zeta function for each positive integer n.
,
where
, s and a are complex variables.
November 07, 2009 |
Finite Series Are Expressed in Terms of n-th Partial Sum of Hurwitz Zeta Function
For real x and each positive integer n,
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October 25, 2009 |
Finite Series in General Form
For real x ≠ 0 and each positive integer n,
September 26, 2009 |
50 Identities of Power Summation (Update)
July 8, 2009 |
A Family of Finite BBP-Type Series in the Base of 729
July 1, 2009 |
Some BBP-Type Series for Computing Pi (Update)
June 14, 2009 |
April 12, 2009 and May 3, 2009 |
Some Finite Series Found in Closed-Form
April 9, 2009 |
December 14, 2008 |
December 13, 2008 |
August 12, 2008 |
The infinite series of the BBP-type formula is found and used to compute the digits of the constant .
August 2, 2008 |
May 1, 2008 |
Power Sum
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Sum of Partial Power Sums
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January 31, 2008 |
<<Previous Next >> |
May 15, 2007 |
The following general series formula is true:
April 07, 2007 |
The fast conergent series are used for computing the logarithm constants log 2 and log 3 (updated).
February 14, 2007 |
December 23, 2006 |
The formula below is true for all .
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.
October 15, 2006 |
Finite Alternative Odd Power Series
September 17, 2006 |
August 25, 2006 |
July 04, 2006 |
May 29, 2006 |
May 28, 2006 |
(This series has been found and can be used to compute directly the n-th digit of ln(7) without computing any prior digits. Notice that the base of this series is 729.)
May 23, 2006 |
May 15, 2006 |
April 21, 2006 |
March 7, 2006 |
March 6, 2006 |
This series can be written in product form:
, which is
.
March 4, 2006 |
January 16, 2006 |
<<Previous Next>> |
December 18, 2005 |
December 07, 2005 |
whereis the Euler constant.
November 12, 2005 |
November 4, 2005 |
where constant e is the base of the natural logarithm function.
whereis Apéry’s constant.
where G is Catalan’s constant.
whereis Riemann zeta function.
August 26, 2005 |
August 26, 2005 |
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