Series Summary is the outlines and brief notes of the series formulas of related posts that have been published on the Series Math Study website.
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,
, where 36x^{3}k^{3}  7xk ± 1 ≠ 0, k = 1,2,3, ... .
It may be rewritten in the form
,
where x < 1/2 or x > 1/2.

August 02, 2012 
A Special Series , where is the digamma function that is defined.

July 29, 2012 
Some Special Series
where , and are the Zeta constants.

June 2, 2012 
A Closed Form of Special Value of Gamma Function , where m is a positive integer.

May 20, 2012 
A Special Limit Expression Involving Gamma Function , where x is real.

January 05, 2012 
New Formula of Gamma function Approximation The Gamma function approximation gives a high level of accuracy for real x,
Notes 1. the natural logarithm of the Gamma function on the left hand side. 2. In mathematics, the Gamma function is an important function that many other special functions depend on. If the closedform of Gamma function is found, the chance of resolving the Riemann hypothesis is very high. Unfortunately, its existence is still not known.


November 06, 2011 
A New Approximation Formula for Computing the Nth Harmonic Number (Update) A new approximate formula of giving more digits of accuracy for computing the nth Harmonic Number is found as follows:
, whereis Euler constant and n is a positive integer.

June 24, 2011 
New Harmonic Number Approximation Formula , whereis Euler constant and n a is positive integer.

January 02, 2011 
A special product series gives
,
where Γ is Gamma function, and Γ(1/3) = 2.6789385347... .



A New Approximation Formula for Computing the Nth Harmonic Number (Update)
A new approximate formula of giving more digits of accuracy for computing the nth Harmonic Number is found as follows:
,
whereis Euler constant and n is a positive integer.
Below is the computation table of some approximate and exact Harmonic Numbers.
n  (Exact Computation)  (Approximate Computation) 

1  1  1.0000364756158384 
2  1.5 (= 3/2)  1.500001060257485 
3  1.8333333333333333 (= 11/6)  1.8333334197475766 
4  2.083333333333333 (= 25/12)  2.0833333459100944 
5  2.283333333333333 (= 137/60)  2.2833333359731323 
6  2.45 (= 49/20)  2.4500000007120852 
7  2.5928571428571425 (= 363/140)  2.5928571430876115 
8  2.7178571428571425 (= 761/280)  2.7178571429427802 
9  2.8289682539682537 (= 7129/2520)  2.828968254003711 
10  2.9289682539682538 (= 7381/2520)  2.928968253984269 
11  3.0198773448773446  3.0198773448851135 
12  3.103210678210678  3.1032106782146784 
13  3.180133755133755  3.1801337551359232 
14  3.251562326562327  3.2515623265635525 
15  3.3182289932289937  3.3182289932297135 
16  3.3807289932289937  3.3807289932294307 
17  3.439552522640758  3.4395525226410317 
18  3.4951080781963135  3.4951080781964894 
19  3.547739657143682  3.547739657143797 
20  3.597739657143682  3.5977396571437588 
50  4.499205338329423  4.499205338329425 
100  5.187377517639621  5.18737751763962 
150  5.591180588643881  5.591180588643878 
200  5.878030948121446  5.8780309481214434 
(November 11, 2011)
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 e^{x}. This formula is also expressed in terms of logarithm to compute complex z.)


October 16, 2010  
Two Series are found in Connection with Mathematical constants,


August 03, 2010  
Two Series in Connection with Mathematical constants


July 03, 2010  
Sums in which the Square Root of Two and Other Constants Appear are Given by


June 27, 2010  
A Surprising Sum in whichAppears is Given by  
.


April 18, 2010  
Some Special Infinite Series  


February 09, 2010  
Sums of Reciprocals of TwoTerm Squares Table  


Sums of Reciprocals of TwoTerm Cubes Table


January 30, 2010  
Series in Limit Form 

Given n is a positive integer. The following series are found in a closed form as n tends to infinity.


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


January 08, 2010  
Two Beautiful Series in Connection with zeta and Pi Constants


January 01, 2010  
A Curious Series in Connection with EulerMascheroni, Pi Constants For positive integer n, , whereis EulerMascheroni constant. The above series can be transformed into another form, namely .
Another curious series is found in connection to Pi .
(Happy New Year 2010) 
Go down deep enough into anything and you will find mathematics. (Dean Schlicter)
We work on exploring and developing math series. Each time a series or a group of series has been developed completely, it is posted on this website. Below are the patterns of some series listed in date order. More related series can be found in the Series Outline section, which consists of Random Series and Series Summary. Random Series is a place where it keeps all math series without classifying to a specific type. Series Summary is a SMS's part in which it keeps the math series in relation to the dates posted on this website.
December 20, 2009 
Infinite Series in Connection with Pi Constant




November 26, 2009 (Happy Thanksgiving) 
Finite Series in Connection with Apéry, Pi Constants
The nth 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 nth Partial Sum of Hurwitz Zeta Function
For real x and each positive integer n,
, x ≠  k,  (k+1).
This finite series is defined in the Hurwitz zeta function form. Read more >>
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 BBPType Series in the Base of 729
July 1, 2009 
Some BBPType Series for Computing Pi (Update)
June 14, 2009 
April 12, 2009 and May 3, 2009 
Some Finite Series Found in ClosedForm
April 9, 2009 
My fellow Americans: ask not what your country can do for you  ask what you can do for your country.  
John F. Kennedy 
We work on exploring and developing math series. Each time a series or a group of series has been developed completely, it is posted on this website. Below are the patterns of some series listed in date order. More related series can be found in the Series Outline Book section, which consists of Random Series and Series Summary. Random Series is a place where it keeps all math series without classifying to a specific type. Series Summary is a SMS's part in which it keeps the math series in relation to the dates posted on this website.
December 14, 2008 
December 13, 2008 
August 12, 2008 
The infinite series of the BBPtype formula is found and used to compute the digits of the constant .
August 2, 2008 
May 1, 2008 
Power Sum

Sum of Partial Power Sums

January 31, 2008 
<<Previous Next >> 
Divided by 0 is often where a transcendental number or an irrational number emerges.  
(T.V.) 
May 15, 2007 
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 
September 17, 2006 
August 25, 2006 
July 04, 2006 
May 29, 2006 
A fast convergent series of BPPtype formulas has been found and can be used for computing the nth digit of in base 4096 without computing any prior digits.
May 28, 2006 
(This series has been found and can be used to compute directly the nth 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 
Note: the author, who created and posted these series on this page, sent 'the idea of these series' to a 'guy' who published the new findings based on Ramanujan's original series for requesting whether those series above are new. There were no answers but ... four months later that 'guy' published in his another article and indicated that the findings in his earlier publish was in the bad direction, and he recomputed his findings based on 'the idea of these series' without mentioning the source of the original inspiration leading to that change!
<<Previous Next>> 
I am interested in mathematics only as a creative art.  
Godfrey Harold Hardy 
December 18, 2005 
December 07, 2005 
November 12, 2005 
November 4, 2005 
August 26, 2005 
August 26, 2005 
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<< Pervious 
Number is the within of all things. (Pythagoras)