Random Series

Random Series is a place where it keeps all math series without classifying to a specific type.  In this Random Series we explore series formulae that have connections to sine, cosine, tan, cot, sinh, cosh, tanh, coth, and special math constants. Since more series formulae are often added into SMS website, the series are not classified to related groups, and they are randomly organized into the multiple sections with dated memos. Most of the series found under this Series Book Outline have already been posted on the old link of SMS website, which is www.seriesmathstudy.com/startsms

 


My God, the mazes must be enormous.  (Kurt Gödel)

16x16 Magic Square

In-Text or Website Citation
Tue N. Vu, 16x16 Magic Square from Series Math Study Resource.

Hyperlink: http://www.seriesmathstudy.com/sms/16x16_magic_square

27x27 Magic Square

In-Text or Website Citation
Tue N. Vu, 27x27 Magic Square, 9/11/13, from Series Math Study Resource.

Hyperlink: http://www.seriesmathstudy.com/sms/content/27x27-magic-square

Sum Of Partial Factorials

Sum Of Partial Factorials
(December 23, 2006)
 
This page shows a general formula called sum of partial factorials that is used to generate many identities such that each of identity is true for all positive integers n.
 
General Formula
, where .
 
The general formula (I) can be rewritten in a product form as follows:
 
, where.
 

 
 
 
 
It is known that
 
 or
 
or so on, where .
 
 
Now we want to find a general formula which is of the form
 
 
 
 
or
 
 
 for all positive integers n.
 
 
Indeed, the general formula (I) or (II) is used to derive our desire. Let consider some values of m.
 

 

, which can be proved by using the method of Mathematical induction.
 
It can easily be checked, for instance, n = 3, both sides are equal to 20.
 
.
 
 
 
.
 
It can easily be checked, for instance, n = 2, both sides are equal to 30.
 
                        .
           
 

 

.
 
 
 

We see that for each positive m, (I) generates a general formula in which it is true for all positive integers n.
 

 

 

or

 

.

 
 

 

 

or

 

.

 
 

 

 

or

 

.

 
 

 

 

or

 

.

 
 

 

 

or

 

.

 

 
 
 

General Formula

 

or its summation notation,

, where .
 
 
We express it in terms of factorial form
 
 
 
or its summation notation,
 
, where m, .
 
 
If we treat m as real x, we get the extended formula in terms of x as follows
 
 
 
 
or its summation notation,
 
            , where .
 
 
 
In addition, we get another interested formula that is based on the above results, namely
 
 
or its expansion form,
 
 
, where
 
 
Example
 
Let n = 4, we obtain a beautiful identity
 
 
for all real x.
 
 
Here are the lists of identities for different values of n. These identities are true for all real x.
 

 

 

or its product notation,

 

.

 
 

 

 

or its product notation,

 

.

 
 

 

 

or

 

.

 

 

 
 

 

or its product notation,  

 

.

 


Other finite series:

 


Number proceeds from unity. (Aristotle)

Repeated Power Sum

Repeated Power Sum
(June 27, 2005)
 
The followings are repeated power sum formulas that are true for all positive integers. The readers can prove them by using principle of Mathematical Induction.   (The way of how the Conjecture of these formulas is established is a different story.   It connects to the work of applying zeta(2) in the purpose of finding the exact formulas for calculating the integrated intensity, power and energy of white-light rays in term of time by letting white light beam with initial intensity * travels in closed-paths so that its initial rays are returned back to the same its initial direction and plane as where it begins. This technique is called an integrated white light method that can be used to build a powerful intensity light source from a small intensity light source. We are still working on it, and will publish this article in the Blog section when done.)
 
We introduce here the new form of some formulas that have been found with coming up a Conjecture in the following:
 
 Assuming that is defined as follows
, where are positive integers.
 
We define two expressions
 
and
,
where is represented for repeated power series.
 
We prove that
or
 
 
Based on our initial assumption and definition, we obtain the repeated power series formulae of for k = 1 to 7 as follows:
 
1.       or or
 
 .
 
Example:
Let k=1 and n = 4, we see both sides of (1) are equal to 20. Indeed, the left hand of (1) is , and the right hand of (1) is
.
     
 
2.       or
 
.
 
Example:
Let k=2 and n = 4, we see both sides of (2) are equal to 50.  Indeed, the left hand of (2) is
, and the right hand of (2) is
            .
 
 
 
3.       or
 
.
 
 
4.       or
 
.
 
 
5.       or
 
.
 
 
6.       or
 
.
 
 
7.       or
 
.
 
Notice,
.
 
 
 
8.             .
 
 
9.       .
 
The expression (9) shows the connection to Euler Constant, Sum of Power and Zeta Series.
 
 
Notice that we have
.
 
Rewrite it in term of summation symbol
 (**)
 
Multiplying both sides of (**) by , we have
 
           .
Therefore,
 
        .
 
We thus obtain a remarkable formula that shows the connection to Euler Constant, Sum of Power, and Zeta function.
,
 
where
 
Or , where A is Glaisher’s constant = 1.28243…
 
 
(June 27, 2005)
 
 

In mathematics when we face a critical problem we cannot solve it with the obvious proof, we understand that the current math tools have failed to reach beyond it. This implies that we need to develop another mathematic tool to solve for it, and this process would  take centuries. (T.V.)
 

Other finite series:

 

A Brief Note - Sum of Riemann Zeta Function - Digamma Function

(June 14, 2009)

In this brief note we show how to derive a sum of Riemann zeta function [1*] in terms ofand Euler constants () with involving the Gamma function [2*].

.

This sum is also used as a connection to Digamma function [3*] in which some special values of Digamma function are computed in closed-forms.


 

Recall that the Riemann zeta function,, is defined 

 (Riemann’s integral form),
 
where  is the Gamma function.
 
 
Euler was the first person to define the Gamma function as a definite integral form
 
             (Euler’s integral form).
 
 
From the Euler’s integral form, it is not difficult to show Gamma function to satisfy the following identities:
 
.
 
            .
 
 
It is known that Euler’s product sine series converges to sine, namely
 
(Euler's product sine).
 
If we replace the variable x by ix, where , we then obtain
 
.
 
 
The Gamma function also satisfies the functional equation
 
.
 
Therefore, we can express the sum of Riemann zeta function in terms of and Euler constants with involving Gamma function by taking the log of both sides of Euler’s product sine series.  We then take the derivative of both sides in associating with the given identities above to obtain the following beautiful series:
 
,
 
where  is Euler's constant and its value is 0.5772156649015328606… .
 
 
Some special values
 
 
.
 
 
 
.
 
 
 
.
 

 
If we set x = -x in (I), we obtain the alternative series
 
.
 
Special values
 
 
 
 

.

 


Digamma Function ()

The digamma function is defined as

.  (II)

 

It is interested that by differentiating (I) and its result gives a connection to digamma function which satisfies the following expression:

. (III)

 

Special values

. (Notice that at x = 1 does not satisfy (III) except its definition (II).)

 

.

 

.

 

.

 

 

.

 

.

 

.

(Notice that we evaluate the special values ofand. Readers should re-verify the accuracy of these special values before usage.)


Relation Series

 


References

[1*] Riemann zeta function

 

[2*] Gamma function

 

[3*] Digamma function

(Note: the above links may be changed by other websites in future.)


More about Riemann, especially Riemann hypothesis, which is part of Problem 8 in Hilbert's list of 23 unsolved problems, and is one of the Clay Mathematics Institute Millennium Prize Problems.

 


In-Text or Website Citation
Tue N. Vu, A Brief Note - Sum of Riemann Zeta Function - Digamma Function, 06/14/2009, from Series Math Study Resource.
Hyperlink: http://www.claymath.org/millennium-problems/millennium-prize-problems.

 A mathematical truth is neither simple nor complicated in itself, it is. (Emile Lemoine)

Alternative Series I

Alternative Series I
 
 
 
1.      .
 
 
2.      .
 
 
 
3.      .
 
 
 
4.        
 
       or
 
      .
 
 
 
5.     
 
or
 
 
 
6.     
 
 
or
 
.
 
 
 
7.      .
 
 
8.      .
 
 
9.      .
 
 
10.  .
 
 
11. 
 
 
or
 
.
 
 
12.  .
 
 
 
13.  .
 
 
14.  .
 
 
15.  .
 
 
16. 
 
 
or
 
.
 
 
17.        
 
 
or
 
.
 
 
 
18. 
 
 
or
 
.
 
 
19. 
 
 
or
 
.
 
 
 
20.   .
 
 
 
21.   .
 
 
22. 
 
 
23. 
 
 
 
24.  .
 
 
25.  .
 
 
26.  .
 
 
27. 
 
 
.
 
Notice,
 
.
 
 
28.  .
 
 
29.  .
 
 
30.  .
 
 
31.   .
 
 
32.  .
 
 
33.  .
 
 
34.  .
 
 
35.  .
 
 
36.  .
 
 
37.  .
 
 
 
38.  .
 
 
39.  .
 
 
40.  .
 
 
 
41.  .
 
 
42.  .
 
 
43.  .
 
 
 
44.  or
 
or .
 
 
45.  .
 
 
46.  .
 
 
47.  .
 
 
48.  .
 
 
49.  .
 
 
50.  .
 
 
51.  .
 

An irrational number ... lies hidden in a cloud of infinity. (Michael Stifel)

Alternative Series II

1.     , where.

 
2.      .
 
 
3.      .
 
 
 
4.      .
 
 
 
     or
 
      
      .
 
 
5.      .
 
 
 
6.      .
 
 
 
7.      .
 
 
 
8.      .
 
 
 
9.      .
 
 
 
 
10.  .
 
 
 
11.  .
 
 
 
 
12.  .
 
 
13.  ,
where
 
                     .
 
                     .
 
                    , etc.
  
 
 
14.  .
 
 
 
15.  .
 
 
 
16.   .
 
 
 
17.     .
 
 
 
18.  .
 
 
 
19.  .
 
 
 
20.  .
 
 
 
21.  .
 
 
 
22.  , where  is Euler Constant (see Math Constants).
 
 
 
23.   .
 
 
 
24.   .
 
 
 
25.  .
 
 
 
26.  .
 
 
 
27.  .
 
 
 
Recall that .
 
 
 
28.  .
 
 
 
29.  .
 
 
 
30.  .
 
 
 
31.  .
 
 
 
32.  .
 
 
 
33.   .
 
 
 
34.  .
 
 
 
35.  .
 
 
 
36.  .
 
 
 
or .
 
 
 
37.  .
 
 
 
38.  .
 
 
 
39.  .
 
 
 
40.  .
 
 
 
41.  .
 
 
 
42.  .
 
 
 
43.  .
 
 
 
44.  .
 
 
 
45.  .
 
 
 
46.  .
 
 
 
47.  .
 
 
 
48.  .
 
 
 
49.  .
 
 
 
50.  , where n is an integer.
(The exact sum of the first n terms of harmonic series can be calculated)
 
 
 
51.  , where .
 
 
 
52.  , where .
 
 
 
53.  .
 

 

Alternative Odd Series

Alternative Odd Series
 
1.       , where
 
2.      
 
3.      
 
4.      
 
5.      
 
6.      
 
7.      
 
8.      
 
9.      
 
10.  
 
11.  
 
12.  
 
13.  
 
14.  
 
15.  
 
16.  
 
17.  
 
18.  
 
19.  
 
20.  
 
21.  
 
22.  
 
23.  
 
24.  
 
25.  
 
26.  
 
27.  
 
28.  
 
29.  
 
30.  
 
31.  
 
 
32.  
 
33.  
 
 
34.  
 
 
35.  
 
 
36.  
 
 
37.   , where
    and
 
 
38.   , where
     and

BBP-Type Series in Connection with one of the Roots of Quartic Equation

December 17, 2008
 
The formulas below show some irrational numbers in connection with the constant Pi.
 
 
, which is one of the roots of the quartic equation
 
.
 
 
 
 
.
 
 
 
 
.

 

All the above BBP-Type series are the non-integer bases.


In-Text or Website Citation
Tue N. Vu, BBP-Type Series in Connection with one of the Roots of Quartic Equation, 12/17/2008, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/irrationalnum.


Well done is better than well said.
Benjamin Franklin

 

Cot and Tan

 
1.     
 
2.     
 
3.     
 
4.     
 

 

Even Riemann Zeta

The Values of Riemann Zeta Function for Positive Even Integers
 
For any positive integer 2n,
,
where is a Bernoulli number.
 
 
1.       , where
 
2.     
 
3.     
 
4.     
 
5.     
 
6.     
 
7.     
 
8.     
 
9.     
 
10. 
 
11. 
 
12. 
 
13. 
 
14. 
 
15. 
 
16. 
 
17. 
 
18. 
 
19. 
 
20. 
 
21. 
 
22. 
 
23. 
 
24. 
 
25. 
 
26. 
 
27. 
 
28. 
 
29. 
 
30. 
 
31.  , where
    and
           
 
 
32.  , where
 
    and
      
     
 
33.  , where
    and

 

Even Zeta in Odd Form.

Finite Alternative Odd Power Sum

 Finite Alternative Odd Power Sum

(October 15, 2006)
 
The sums of the following expressions are true for any positive integer n. 
 
1.      .
 
Example:
            Let n = 3, the left-hand side of the above expression is 1-3+5 = 3 and its right-hand side is –3(-1) = 3. 
 
 
2.      .
 
Example:
            Let n = 3, the left-hand side of the above expression is 1-9+25 = 17 and its right-hand side is –2(9)(-1) –1 = 17. 
 
 
3.      .
 
Example:
            Let n = 4, the left-hand side of the above expression is 1-27+125-343 = -244 and its right-hand side is [3(4) – 4(64)](1) = -244. 
 
 
4.      .
 
 
5.      .
 
6.      .
 
 
7.      .
 
8.      .
 
 
9.      .
 
 
10.  .
 
 
11.  .
 
 
12. 
.
 
 
13. 
.
           
 
14. 
.
 
 
15. 
.
 
 
16. 
.
 
 
17. 
.
 
 
18. 
                       .
 
 

 

Other finite series:


In-Text or Website Citation
Tue N. Vu, Finite Alternative Odd Power Sum, 10/15/2006, from Series Math Study Resource.
Hyperlink: http://www.seriesmathstudy.com/finitealternativeodd.

 


A mathematical truth is neither simple nor complicated in itself, it is.  
 Emile Lemoine  

Power Sum

Power Sum 

 For each positive integer n,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


More >> (50 identities of Power Sum in text format)

Download (powersum.txt)


 

Other finite series:

 

Power Sum (Text Format)

50 Identities of Power Summation

(09/26/2009 - update)

For each positive integer n,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(09/26/2009 - Update)



In-Text or Website Citation
Tue N. Vu, 50 Identities of Power Summation, 09/26/2009 (update), from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/powersumtextformat.

We all agree that your theory is crazy, but is it crazy enough?
                                                                        Niels Bohr (1885-1962)

Finite Odd Power Sum

Finite Odd Power Sum
 
 
 For each positive integer n,
 
1.      .
 
2.      .
 
3.      .
 
 
4.    14 + 34 + 54 + ... + (2n-1)4 = (16/5)n - (8/3)n3 +  (7/15)n.
 
5.      .
 
 
6.    16 + 36 + 56 + ... + (2n-1)6 = (64/7)n7 - 16n5 +  (28/3)n3  - (31/21)n .
 
7.      .
 
8.   18 + 38 + 58 + ... + (2n-1)8 =  (256/9)n9 - (256/3)n7 + (1568/15)n5 - (496/9)n3 + (127/15)n .
 
9.      .

 

10.  110 + 310 + 510 + ... + (2n-1)10 = (1024/11)n11 - (1280/3)n9 + 896n7 - 992n5 + 508n3 - (2555/33)n .

 

11.  111 + 311 + 511 + ... + (2n-1)11 = (512/3)n12 - (2816/3)n10 + 2464n - (10912/3)n6 + 2794n4 - (2555/3)n2 .

 

12.  112 + 312 + 512 + ... + (2n-1)12 = (4096/13)n13 - (2048)n11 + (19712/3)n9 - (87296/7)n+ (67056/5)n- (20440/3)n3 + (1414477/1365)n .

 

13.  113 + 313 + 513 + ... + (2n-1)13 =(4096/7)n14 - (13312/3)n12 + (256256/15)n10 - (283712/7)n8 + (290576/5)n6 - (132860/3)n4 + (1414477/105)n2 .

 

14.  114 + 314 + 514 + ... + (2n-1)14 = (16384/15)n15 - (28672/3)n13 + (652288/15)n11 - (1134848/9)n9 + (1162304/5)n7 - (744016/3)n5 + (5657908/45)n3 - (57337/3)n.

 

15.  115 + 315 + 515 + ... + (2n-1)15 = 2048n16 - 20480n14 + (326144/3)n12 - (1134848/3)n10+ 871728n8 - (3720080/3)n6 + (2828954/3)n4 - 286685n2 .


 

Other finite series:

 

The Reciprocal of Infinite Product

The Reciprocal of Infinite Product

 

The reciprocal of the beautiful infinite product of nested radicals due to Vieta in 1592 is decomposed into partial fractions of the infinite series as shown below.

            , in which

.

 

 

 


The mathematical sciences particularly exhibit order, symmetry,

and limitation; and these are the greatest forms of the beautiful.

                                                                                                                                                  Aristotle

 

Series List I

June 25, 2005
 
 
 
 
 
 

 

or
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
or
 
 
 
 
 

God does not care about our mathematical difficulties. He integrates empirically. (Albert Einstein)

Finite Series are Expressed in Terms of N-th Partial Sum of Hurwitz Zeta Function

Brief Note - Introduction a generalization form of n-partial sum of Hurwitz Zeta Function.

 

The n-th partial sums of the expressions (I) and (II) below are found in closed-form for each positive integer n and real x ≠ -k and x ≠ -(k+1).

 

.

 

 .

 

Special values

 

.

 

Let x = 0,

.

 

. (Correct - 11/17/2009)

 

At x = 1,

 

.

 

Example 

 

.

 

A Generalization Form of n-th Partial Sum of Hurwitz Zeta Function

Recall that Hurwitz Zeta Function [1] is defined for complex arguments s and a by

 , where Re(s) > 1 and Re(a) > 0.

 

We now define a new generalization form of n-th partial sum of Hurwitz zeta function above as follows:

.

 

Based on this new define, (I) and (II) are then rewritten in terms of n-th partial sum of Hurwitz zeta function for Re[a] = x, R[s] = 2, and each positive integer n as shown in the following:

(III)     .

 

(IV)    .

 

The identities (III) and (IV) are also true for complex number a by replacing x = a.

(November 07, 2009)

(Update formula syntax and definition - December 05, 2009)


Other related series

50 Identities of Power Summation


References

[1]  http://en.wikipedia.org/wiki/Hurwitz_zeta_function


 In-Text or Website Citation
Tue N. Vu, Finite Series are Expressed in Terms of N-th Partial Sum of Hurwitz Zeta Function, 11/07/2009, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/nov_07_2009.

Mathematicians might have only acknowledged the beauty of mathematical formulas silently after understanding them.

The exquisiteness on human faces is an ephemeral beauty but the charming manifestation of mathematics is a beauty in all eras. (T.V.)

Finite Series in Connection with Apéry, Pi Constant

The n-th partial sums below are true for each positive integer n.

 

 

 

 

The notationsandin (I) and (II) represent the special values of a new generic formula that we define it as an extensive notation from Hurwitz zeta function [1*] for n-th partial sum as follows 

,

where s and a are complex variables with Re(s) > 1 and Re(a) > 0.

 

When n tends to infinity, the extensive notation [2*] is then expressed as

.

 

Special Case

In the limit as n approaches to infinity, both series (I) and (II) converge and its values are

 

, where is Apéry's constant.

 

.

 

Example

 

 .

 

 (November 26, 2009 - Happy Thanksgiving)


Question:

For any positive integers m1, m2 and m3,

 

Yes. 

 


 Other related series

Finite Series are Expressed in Terms of N-th Partial Sum of Hurwitz Zeta Function

50 Identities of Power Summation

 


References

[1*] Hurwitz function, en.wikipedia.org/wiki/Hurwitz_zeta_function, from Wikipedia resource.

[2*] The purpose of our website is to show a beauty of series.  We introduce a new extension of the notation of Hurwitz zeta function for n-th partial sum because there exist such series (I) and (II).  We share our results on the internet.  The acceptance of this notation is the work of other men.

 


In-Text or Website Citation
Tue N. Vu, Finite Series in Connection with Apéry, Pi Constants, 11/26/2009, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/nov_26_2009.
 

 God speaks to us in many ways. Math is one of them. (T.V.)

 

 

Finite Series in General Form

For real x ≠ 0 and each positive n,

 

 

 

 

(October 25, 2009)


In-Text or Website Citation
Tue N. Vu, Finite Series in General Form, 10/25/2009, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/seriesunknownname1.

God created the integers. (Stephen Hawking) 

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 17, 2010) 
 

 

We show a beautiful art of a generic infinite series that links to three special sequences and the math constant(Pi) via integers only.

 

For any positive integer

,

where

,

,

and

.

 

The formula (I) is a generic infinite series expressed in terms of positive integer n in conjunction with three special expressions,, and.  The expressions,, and, which are Binet's formula-like [*1], represent three special sequences for n = 1, 2, 3, ... as shown in Table -1.  For each value of n, (I) gives a series of a closed form that contains Pi and other rational numbers.

 

 

 

The interesting point is that the sequences generated fromandare related to Diophantine property, and they can be found in the On-Line Encyclopedia of Integer Sequences site [*1] for id:A028230 and id:A103772, respectively.  We observe that the ratio of any consecutive numbers of these three sequences, for instance, 5852 / 420 or 564719 / 40545, converges to the constant 13.9282032... as n grows large. The exact value of 13.9282032... is determined in a closed-form, namely

 

 

Example

Below we illustrate the outcomes of (I) when considering n = 1, 2 and 3.
 

.

Or this series is written with the index k starts at 0,

.

 

.

If you use symbolic math software to compute the 32 terms of the following expression

,

you will obtain this series,

.

 

.

 

Conclusion and Comments

For each value of integer n, the generic formula (I) always gives a series in a closed form that contains a single Pi and other rational numbers.  The appearance of the constant Pi in each of these series implies us a means to compute the decimal digits of Pi.  However, the formula (I) is a slow convergent series.  Therefore, the family of these series of (I) is not an ideal formula that can be used to calculate billion decimal digits of Pi.  Reader may see that when n approaches to infinity, both sides of (I) approach to zero while its index k is still stepping through infinity.  The beauty of (I) is the outcomes of rational form associating with a single Pi to those special sequences defined in the expressions,andso that they can harmonize with the constant Pi through integers only.  In addition, if reader wants to use (I) to generate series for large values of n, need to replace all expressions,andby its recurrence formulas [*2] such that all square roots of 3 from these expressions are removed.

 

Relationship among Three Special Sequences,and.


Notes

[*1] Diophantine property, www.research.att.com/~njas/sequences/Seis.html, Encyclopedia of Integer Sequences resource. The reference links in this webpage may be changed by other websites in future.

[*2] Use the given ids, A028230 and id:A103772, to search for the recurrence formulas oforfrom On-Line Encyclopedia of Integer Sequences site.  Click here to quickly view it.  We also found the recurrence formula ofis defined as follows.

a(n) = 14a(n-1) - a(n-2)

a(0) = 1

a(1) = 2

(Recall that (I) is true for.    Click here see the Relationship among,and.)


In-Text or Website Citation
Tue N. Vu, A Generic Infinite Series Found Linking Three Special Sequences, 01/17/2010, from Series Math Study Resource.
Hyperlink: http://www.seriesmathstudy.com/sms/jan17_2010_link.

As far as the laws of mathematics refer to reality, they are not certain;
and as far as they are certain, they do not refer to reality. (Albert Einstein)

 

 

Series List II

July 5, 2005
 
1.     
 
 
2.     
 
 
3.     
 
 
4.     
 
 
5.     
 
 
6.     
 
 
7.     
 
 
8.     
 
 
9.     
 

As for everything else, so for a mathematical theory: beauty can be perceived but not explained.  
 Arthur Cayley  

Some Series in Connection with Pi Constants

Some Series in Connection with Pi Constants 

 

  

 

 

 

  • .

(December 20, 2009)

 

Some Special Infinite Series

 

 

 

 

 

 

 

 

 

 


 In-Text or Website Citation

Tue N. Vu, Some Special Infinite Series, 04/18/2010, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/april18_2010 .
 

Perfect numbers like perfect men are very rare. (René Descartes)
 

Series List III

July 9, 2005
 
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Series List IV

(November 12, 2005)
1.     
 
2.     
 
3.     
 
4.     
 
5.     
 
6.     
 
7.     
 
8.     
 
9.     
 
10. 

God does not care about our mathematical difficulties. He integrates empirically. (Albert Einstein)

Series List V

(December 19, 2005)

 

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  •          (product series)

 

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  •         
 

(= 0.577215664901532…, which is the Euler constant.)


In-Text or Website Citation
Tue N. Vu, Series List V, 12/19/2005, from Series Math Study Resource.
Hyperlink: http://www.seriesmathstudy.com/series5.

Mathematics is the supreme judge; from its decisions there is no appeal.

                                                                                                             Tobias Dantzig

Series List VI

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Return SMS

Number is the within of all thing. (Pythagoras)

Pi

*(Pi)
 
The constant pi, denoted*, is a real number and found everywhere in nature.  The constant pi is known to be irrational and transcendental number. The precise pi is determined through the ages and its decimal is never ending.  In nowadays the value of pi is known to more than one trillion places via the methods of the Gregory-Leibnize series, Maclaurin series and Machin-like formulas.  Most recently, the new formula called BBP-type series is used to compute the nth binary or hexadecimal digit of pi without having to compute the preceding n –1 digits.  Below are the BBP-type series with the base of 16 and 4096.

 

The formulas for the BBP-type with the base of 16 are
 
  •  .   (BBP-Type - this series was discovered by Bailey-Borwein-Plouffe in 1995.)
 
 

 

  • .

 

 
The BBP-type series formulas are found in the base of 4096 that causes the infinite series converges extraordinarily rapidly, namely

 

  •  .

 (posted May 29, 2006)

 
  • .

(posted July 4, 2006)

 
(Click here to view these series in full window size without the left/right menu bar.) 

 
Other infinite series of pi are found as showing in the following:
 
  •  

 

  

 

.

 
or
  •    

 

 

.

 
 
 
  •    

 

 

 

 

 

 

 

.

 
(June 20, 2005)
 
 
 
 
 
 
 
 
 
 
 
 
.
 
 
 
  • .

 

(05/29/2006)
(07/04/2006 - update)
(03/17/2009 - update)

 

More Related Series 

 

Main SMS


In-Text or Website Citation
Tue N. Vu, Pi, 07/04/2006, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/pi1.

The mathematical rules of the universe are visible to men in the form of beauty.

(John Michel)

 

A Brief Note on Nth Partial Sum of Harmonic Series

A Brief Note on Nth Partial Sum of Harmonic Series

(09/05/2006) 

The purpose is to consider some series in connection with harmonic series and establish expressions in recurrence relation to harmonic number.
 

Some Series in Connection with Harmonic Series
 
The harmonic series is defined as the sum of 1, 1/2, 1/3, …, and it is written in expanded form with nth partial summation notation of harmonic series as follows:
            
            
 
 Its sum diverges to infinity as n tends to infinity, namely
 

.

 

The alternating harmonic is defined as the sum of 1, -1/2, 1/3, -1/4, … . Its sum converges to ln (2), namely
 
.
 
Rewrite the alternating series in the form of even and odd harmonic series as follows:
 
.
 
Therefore, the odd harmonic series also diverges to infinity.
 
.
 
The odd harmonic series is rewritten in another form as shown in the following steps:
 
 
 
                  
 
    
 
    
 
Therefore,
 
 .
 
 
Now we see that the harmonic series is transformed into the form of
 
.
 
However, if the k term from the nominator of the above expression is removed, we then obtain
 
.
 
If we replace 1/[ (2k-1)2 - 1/22 ] by 1/[ (2k-1)2  + 1/22], its sum of alternating series converges, namely
 
.
 
 
 

  
Expressions in Recurrence Relation to Harmonic Number
 
Define
 
 
  is called harmonic number. (Notice for different notation in Wiki.)
 
 
We then obtain the following recurrence relation:
 
           
 
However, if we do the recursive substitution for , it gives a simple relationship between , namely
 
 
    
 
    
 
    
 
It is also easy to see that
 
.
 
 
Thus, we can state harmonic number as follows:
  
 
(09/05/2006)

Constant ln(2)

A New Formula For Computing Constant ln(2)
 
 
The following new infinite series is discovered and used to compute the constant ln(2) to billions of decimal places.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
.
 
Notice that the base of above infinite series is
 
, where k=0,1,2,3,…
 
 
(August 1, 2006)
 


In-Text or Website Citation
Tue N. Vu, A New Formula For Computing Constant ln(2), 08/01/2008, from Series Math Study Resource.
Hyperlink: http://www.seriesmathstudy.com/log2.

 


It is not every question that deserves an answer.
Publilius Syrus

 

Constant ln(3)

The New Formula For Computing Constant ln(3)
 
 
The following new infinite series is used to compute the constant ln(3) to billions of decimal places.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Notice:
 
, where k=0,1,2,3,…
 
 
(August 6, 2007)
 


In-Text or Website Citation
Tue N. Vu, A New Formula For Computing Constant ln(3), 08/06/2008, from Series Math Study Resource.
Hyperlink: http://www.seriesmathstudy.com/log3.

 


The infinite!  No other question has ever moved so profoundly the spirit of man.  
 David Hilbert  

Finite BBP-Type Series

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

(July 8, 2009)

 

A family of BBP-Type series in the base of 729 is found in terms of a special pattern of finite form such that it generates many other similar series.

Each BBP-Type Series is generated for each positive integer n, namely

.

 


Special Values

As n = 1, the above finite series gives

.

As n tends to infinity, we obtain a convergent series, which is

.


Other BBP-Type Series

Main SMS 


In-Text or Website Citation
Tue N. Vu, Finite BBP-Type Series, 07/08/2009, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/finitebbptypeseries.

Mathematics is a path to display the authentic beauty of nature.

                                                                                                          (T.V.)

Machin-Type Formula

Machin-Type Formula

(April 09, 2009)

 

Some finite series help to find a family of Machin-type formula.

Entry 1.

For each positive integer n,
.
 

(I) illustrates that for each value of n, it gives a Machin-like formula. Therefore, many Machin-like formulas of this similar structure type are generated as n tend to infinity. This establishes a family of Machin-like formula that is depicted in the following example.
 

 

Example.

1. For n = 0, (I) gives
 

.
 


2. For n = 1, (I) gives
 

.

 

3. For n = 2, (I) gives
 

.

 

4. For n = 3, (I) gives
 

.
 

And so on.

 

Entry 2.

Another beautiful identity is used to generate many Machin-type formulas of a class of identities involving a well known formula that Hwang Chien-Lih (1997) used to compute the digits of Pi [1], namely 

 

For each positive integer n,


 

.  

 

Many Machin-like formulas of (II) are generated as n = 0, 1, 2, 3, ..., infinity. 


Collection of Machin-Like formulas 


References

[1]  Chien-Lih Hwang, More Machin-Type Identities, The Mathematical Gazette, 1997, p. 120-121.
.
 


In-Text or Website Citation
Tue N. Vu, Machin-Type Formula, 04/09/2009, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/machintypetv.
 

There are no secrets to success. It is the result of preparation, hard work, and learning from failure.
Colin Power

 

Other Series

 Other Series
 
We define
 
, where are integers > 0.
 
Setting
 
, then
 
 or
 
.
 
Its summation notation is
 
.
 
Multiplying both sides of above equation by ,
 
.
 
Therefore,
 
.
 
 
We thus obtain a remarkable formula that shows the connection to Euler Constant, Sum of Power, and Zeta function.
 
, where .
    
      Or
 
, where A is Glaisher’s constant = 1.28243…
 

Power Sum and Sum of Partial Power Sums

Power Sum and Sum of Partial Power Sums
(May 1, 2008)
 
Power Sum
 
     
      where p, n are positive integers.
 
 
Computing the Power Sum for p = 1, 2, …, 14 gives
 
1.     
    
 
 
 
2.     
     
 
 
 
3.     
      
 
 
 
4.     
      
 
 
 
5.     
     
 
 
6.     
       
 
 
 
7.     
     
 
 
 
8.     
     
 
 
9.     
      
 
 
10. 
       
 
 
 
11. 
      
 
 
12. 
 
      
 
 
                       
 
13. 
      
 
 
 
 
14. 
       
 
 
 
 
Sum of Partial Power Sum (Update of Repeated Power Series, June 27, 2005)
 
Define
,
where m, n, k, and i are positive integers.
 
 
Computing the Sum of Partial Power Sum for m = 1, 2, …, 12 and for all positive n gives
 
1.     
 
 
 
2.     
 
 
 
3.     
 
 
 
4.     
 
 
5.     
 
 
 
6.     
 
 
 
7.     
 
 
 
8.     
 
 
 
1.     
 
 
2.     
 
 
 
 
3.     
 
 
 
 
4.     
 
 
 
(May 1, 2008)
 

Similar seriesFinite Alternative Odd Power Series.

In-Text or Website Citation
Tue N. Vu, Power Sum and Sum of Partial Power Sums, 05/01/2008, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/PowerSumandSumofPowerSum.

Numbers are the highest degree of knowledge. It is knowledge itself.  
 Plato  

Product Series

Product Series
 
 
 
 
 
1.      
 
2.      
 
3.      
 
4.      
 
5.      
 
6.      
 
7.      
 
8.      
 
9.      
 
10.  
 
11.  
 
12.  
 
 
13.  
 
14.  
 
15.  
 
16.  
 
17.   or
 
18.  
 
 
Proofs
 

Some BBP-Type Series

Some BBP-Type Series
 
 
The followings are the BBP-type formulas in which some math constants appear. Some of these series can be used to compute the constant.
 
 
 
 
 
 
 
 
 
 
 
(August 12, 2008)
 


A journey of a thousand miles must begin with a single step.
Lao Tzu

 

Some Finite Series Found in Closed-Form

(April 12, 2009 and May 3, 2009)

 For each positive integer n, the following nth partial sums are found in a closed-form.

 

 

 

 

 

 

 

.  Also, there exists another series with a different sign of k, namely

 

.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This finite series is a beautiful identity in which both sides are equal for each positive integer n.  It converges to 25/576 as n tends to infinity. In addition, this series can be transformed into another form, namely 

 

 

or

 

.

 


Other finite series:

Main SMS


In-Text or Website Citation
Tue N. Vu, Some Finite Series Found in Closed-Form, 04/12/2009, from Series Math Study Resource.
Hyperlink: http://seriesmathstudy.com/sms/finiteseriescloseform1.

 

Success is not final. Failure is not fatal. The courage to continue is what counts.

Winston Churchill

Trig/Hyper Functions

Trigonometric and Hyperbolic Series
 
The below series have connections to sin, cos, sinh, cosh, tanh, coth, and other constants.
 
 
 
 
 
 
Rewrite the above series in summation notation
 
or
 
 
 
 
 
 
or
or

 

Trigonometric Sum

Trigonometric Sum Series
 
 
1.        .
 
 
2.      .
 
 
3.      .
 
 
4.      .
 
 
5.      .
 
 
6.      .
 
 
7.      .
 
 
8.      .
 
 
9.      .
 
 
10.  .
 
 
11.  .
 
 
12.  .
 
 
13.  .
 
 
14.  .
 
 
15.  , where a is real number.
 
 
16.  .
 
 
17.  .
 
 

 

Unknown Name Series

May 15, 2007
 
 
We rewrite the above formula in term of ln and arctanh functions as follows:
 
 
The following expressions are not true but it is displayed here as a question of great interest. Namely, if , then (I) is broken into two sub-expressions: