Series Summary

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.

Main SMS (2012)

Brief Notice - We are no longer going to support this website after 2012. Why?  All the information on this website will be kept as it 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,  

 

,

where 36x3k3 - 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 closed-form of

Gamma function is found, the chance of resolving the Riemann hypothesis is very high.  Unfortunately, its existence is still not known.

 

View this formula in MathML for XHTML editing

 


I do not know how and when the numbers are created.   I only know that numbers are already made available in the structures of nature in which each set number being associated with a structure represents a cold beauty in such a way that has been profoundly designed and eternally engraved in timeless patterns. (T.V.

MAIN SMS (2011)

Brief Notice - This website is going to be temporarily closed to reserve for improvement, and this work probably takes long.  We apologize for not sharing series formulas regularly.
 
 


 
November 06, 2011

A New Approximation Formula for Computing the N-th Harmonic Number  (Update)

A new approximate formula of giving more digits of accuracy for computing the n-th Harmonic Number is found as follows:

 

,

whereis Euler constant and n is a positive integer.

 More >>

 

June 24, 2011

New Harmonic Number Approximation Formula 

,

whereis Euler constant and n a is positive integer.

Read more >>

 

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

I do not know how and when the numbers are created.   I only know that numbers are already made available in the structures of nature in which each set number being associated with a structure represents a cold beauty in such a way that has been profoundly designed and eternally engraved in timeless patterns. (T.V.

A New Approximate Formula for Computing the N-th Harmonic Number

A New Approximation Formula for Computing the N-th Harmonic Number (Update)

A new approximate formula of giving more digits of accuracy for computing the n-th 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.


Table - Computation of Exact and Approximate Harmonic Numbers

n  (Exact Computation)  (Approximate Computation)
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)

In-Text or Website Citation
Tue N. Vu, A New Approximation Formula for Computing the N-th Harmonic Number (Update), 11/11/2011, from Series Math Study Resource.
Hyperlink: http://www.seriesmathstudy.com/sms/ApproximateFormulaHarmonicNum.

Main SMS (2010)

The followings are outlines and brief notes of the series formulas that have been developed and found recently.
 
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.)

 

October 16, 2010

 Two Series are found in Connection with Mathematical constants,

read more >>

 

August 03, 2010

 Two Series in Connection with Mathematical constants

Read more >>

 

July 03, 2010

Sums in which the Square Root of Two and Other Constants Appear are Given by 

 Read more >>

 

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

.

 

April 18, 2010
Some Special Infinite Series
  • .

Read More >>

 

February 09, 2010
 Sums of Reciprocals of Two-Term Squares Table

 

Read more >>

 

Sums of Reciprocals of Two-Term Cubes Table

Read more >>

 

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.

 

  • .

  • .

Read more >>

 

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

Read More >>

 

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

Read more >>

 

January 01, 2010

A Curious Series in Connection with Euler-Mascheroni, Pi Constants 

For positive integer n,

, whereis Euler-Mascheroni constant.

The above series can be transformed into another form, namely

.

 

Another curious series is found in connection to Pi

.

 

(Happy New Year 2010)

 

<<Previous      Next >>


Go down deep enough into anything and you will find mathematics. (Dean Schlicter

Main SMS (2009)

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 SummaryRandom 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 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.  

More >>


 

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, 

,   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, 

.

More >>

 


September 26, 2009

 

50 Identities of Power Summation (Update)

 

July 8, 2009

 

A Family of Finite BBP-Type Series in the Base of 729

 For each positive integer n,
 
.
More >>
 

July 1, 2009

 

Some BBP-Type Series for Computing Pi (Update)

 

More >>

 


 

June 14, 2009
 
 
A Brief Note of the Sum of Riemann Zeta Function and the Digamma Function
The sum of Riemann zeta function,, is found in the closed-form.
 
.

 More >>

 


 

April 12, 2009 and May 3, 2009

 

Some Finite Series Found in Closed-Form

For each positive integer n, the following finite series are found in a closed-form.
 

 

  • . (correct)

  • .

  • .

 More >>


 

April 9, 2009
 
 
Some Finite Series Help to Find a Family of Machin-Type Formula
 
For each positive integer n,
.
 
More >>
 

<<Previous             Next >>


My fellow Americans: ask not what your country can do for you - ask what you can do for your country.  
 John F. Kennedy  

Main SMS (2008)

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 SummaryRandom 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

 
 
An infinite series has a connection with one of the roots of the quartic equation and the constant Pi.
 
,
where  
, which is one of the roots of the quartic equation
 
                     .
 
More >>

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.

  
 
August 12, 2008

 

  The infinite series of the BBP-type formula is found and used to compute the digits of the constant .

 
.
 
Another series is found in terms of other math constants, namely
 
.
 
Click here to see other similar series of this type.
 

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.
 



May 1, 2008
 
 
Power Sum and Sum of Partial Power Sums for any positive integer n. 
 
Power Sum
 
      
                      
  
 
Sum of Partial Power Sums
 
                  
Click here to see other similar formulas.
   

January 31, 2008

 

 
 
.
 
  

  <<Previous    Next >>  

Divided by 0 is often where a transcendental number or an irrational number emerges.  
(T.V.)  

Main SMS (2007)

May 15, 2007

 

 The general series formula below is true for all |x|a and a1, namely
 
.
 
A special case as a = 1 and x = 0, it gives
 
.
 
Click here to see other form.
  

 
April 07, 2007

 

  The fast conergent series are used for computing the logarithm constants log 2 and log 3 (updated).

  

February 14, 2007

 

 

.
 
 (Notice are the special values of the Riemann zeta function at positive integers.)

 

<<Previous  Next>>

Main SMS (2006)

December 23, 2006

 

  The formula below is true for all .

 
.
 
Click here to see more formulas and examples.
 
 

 
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
 
The sums of the following identities are true for any positive integer n.
 

 

 

 

 

More >>

 


September 17, 2006
 
 

 

 
 

August 25, 2006
 
 
GraphFunc is an online program for drawing graphs of basic mathematical functions in 2D and 3D coordinate systems.  Click here to use this tool.
 
 
(August 1, 2006)

 

 
 
 

July 04, 2006
 
 

 
(Twin series – when comparing this series and the one shown below)
 
Some series in relation to pi
 

May 29, 2006
 

 

A fast convergent series of BPP-type formulas has been found and can be used for computing the n-th digit of in base 4096 without computing any prior digits. 

(We do not know whether this series is new)
 
Some series in relation to pi.  
 

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

 

 

 More >>

 
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  

Main SMS (2005)

 


 

December 18, 2005
 
 

 

 
 

December 07, 2005
 
 
 
 

 
November 12, 2005
 
 
 
 
 
 

 
November 4, 2005
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
August 26, 2005
 
 
(This series can be used to compute directly the n-th digit of ln(2) in base 6561 without computing any prior digits)
 
 
 

August 26, 2005
 
 
(This series can be used to compute directly the n-th digit of ln(3) in base 256 without computing any prior digits)
 
 
 
 
 
 
For each positive integer n,
 

 

 
For each positive integer n,
 

.


 

.

The above nth partial sum gives three possible answers: 0, 1 or 1/2 in which its result depends on the starting value of the index k that is a key path to make this sum deterministic. 
 
More >>
 
 

 
 
 
 

 

 
 
 
 .
 

 

 

.

 
 
 

.

 
 

  << Pervious  

 


Number is the within of all things. (Pythagoras)