# 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 . . , whereis the Euler constant. , 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.

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. 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... .

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.

# 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,

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 Two-Term Squares Table

Sums of Reciprocals of Two-Term 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 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)

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.

 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,

.

 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,

.

 July 1, 2009

Some BBP-Type Series for Computing Pi (Update)

• .

• .

 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.

.

 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) . .

 April 9, 2009

Some Finite Series Help to Find a Family of Machin-Type Formula

For each positive integer n,
.

 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

.

 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

.

 May 1, 2008

Power Sum and Sum of Partial Power Sums for any positive integer n.
 Power Sum Sum of Partial Power Sums

 January 31, 2008

.

 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

.

 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.)

# Main SMS (2006)

 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

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)
•

• The new fast convergent series is found for computing the constant log 2.

 July 04, 2006

(Twin series – when comparing this series and the one shown below)

 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)

 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

• or it can be written in product form:

•  .

 March 6, 2006

•  .

This series can be written in product form:

, which is

 March 4, 2006
•

More >>

 January 16, 2006

•  (new?)

• (new?)

• (new?)

• (a beautiful series.  This series can be found in Ramanujan’s second notebook)

•

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!

 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

• , whereis Riemann zeta function.

• , whereis Riemann zeta function.

• .

• .

 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,

.

• A strange sum (Update Nov 02, 2008)

.

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.

• .

.

.

.

• .

Number is the within of all things. (Pythagoras)