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)
| January 05, 2012 |
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New Formula of Gamma function Approximation The new formula of Gamma function approximation for real x has been found in which it provides a high level of accuracy.
(Note the natural logarithm of the Gamma function on the left hand side.) |
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MAIN SMS (2011)
| November 06, 2011 |
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A New Approximation Formula for Computing the N-th Harmonic Number A new approximate formula of giving more digits of accuracy for computing the n-th Harmonic Number is found as follows:
where
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| June 24, 2011 |
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New Harmonic Number Approximation Formula
where
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| January 02, 2011 |
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A special product series gives
 ,
where Γ is Gamma function, and Γ(1/3) = 2.6789385347... .
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,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:
,
where
is 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)
Main SMS (2010)
| October 19, 2010 | ||||
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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 | ||||
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Two Series are found in Connection with Mathematical constants,
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| August 03, 2010 | ||||
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Two Series in Connection with Mathematical constants
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| July 03, 2010 | ||||
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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 Appears is Given by |
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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 | ||||
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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 | ||||
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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  and Pi Constants
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| January 01, 2010 | ||||
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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) |
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.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 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
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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.
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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,
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| 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)
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| June 14, 2009 |
, is found in the closed-form.
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| April 12, 2009 and May 3, 2009 |
Some Finite Series Found in Closed-Form
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| 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 |
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 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 |
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, which is one of the roots of the quartic equation
.| December 13, 2008 |
is a Riemann zeta | August 12, 2008 |
The infinite series of the BBP-type formula is found and used to compute the digits of the constant
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.| August 2, 2008 |
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| May 1, 2008 |
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Power Sum
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Sum of Partial Power Sums
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| 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 |
a and a
1, namely
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.| April 07, 2007 |
The fast conergent series are used for computing the logarithm constants log 2 and log 3 (updated).
| February 14, 2007 |
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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 
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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.
| October 15, 2006 |
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| September 17, 2006 |
| August 25, 2006 |
- The new fast convergent series is found for computing the constant log 2.
| July 04, 2006 |
| 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.
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| May 28, 2006 |
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(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:
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| March 6, 2006 |
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This series can be written in product form:
, which is
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| March 4, 2006 |
, where 
is the Euler constant.
| January 16, 2006 |
(new?), where 
is Apéry’s constant.
(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!
| <<Previous Next>> |
| I am interested in mathematics only as a creative art. | |
| Godfrey Harold Hardy |
Main SMS (2005)
| December 18, 2005 |
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| December 07, 2005 |
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, where
is the Euler constant.
| November 12, 2005 |
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| November 4, 2005 |
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, where
is Riemann zeta function.
, where -
, where
is Apéry’s constant.
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, where G is Catalan’s constant.
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, whereis Riemann zeta function.
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| August 26, 2005 |
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- A strange sum (Update Nov 02, 2008)
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| << Pervious |
Number is the within of all things. (Pythagoras)