16x16 Magic Square

In-Text or Website Citation

Tue N. Vu, 16x16 Magic Square, from Series Math Study Resource. 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)

We present a general formula called as the 'sum of partial factorials'.

General Formula

for m and n ∈ N. Formula (I) holds true for all positive integers m and n. The term in the formula can be expressed in expanded form as:

for m and n ∈ N.

It is known that there exist finite series of the form:

and so on

for.

We endeavor to identify other formulas with a similar pattern to the one described above by studying the characteristics of the subsequent finite series through the method of induction:

or express in the sum notation

or

and so on.

The above pattern can be generalized as:

or

where m and n are positive integers.

Furthermore, it can be expressed in the factorial form:

or

for m and

.

We discover that the sum still holds when we treat m as a real variable x,

or

for x ∈ R and n ∈ N.

Finally, we rewrite it in the summation and production notations as:

for x ∈ R and n ∈ N.

Examine various values of n

n = 2:

n = 3:

n = 4

Later, we note that any forms of the finite series can be obtained by applying the standard formulas for sums of integers powers.

Other finite series:

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

From our preliminary assumption and definition, we derive the repeated power series formulae of

for k = 1 to 7 as follows:

(1)

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 the summation notation,

(**).

Multiplying both sides of (**) by

, which gives

.

Therefore, take sum of both sides from k = 1 to infinity is given by:

.

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-Text or Website Citation

Tue N. Vu, Repeated Power Sum, 06/27/2005, from Series Math Study Resource.

Hyperlink: http://seriesmathstudy.com/sms/repeatpowerseries.

Other finite series:

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

(June 14, 2009)

In this brief note, we demonstrate how to derive the following sum that involves the Riemann zeta function [1*] in terms of π, Euler constants γ, and the Gamma function [2*].

.

This sum is also connected to the Digamma function [3*], where certain special values of the Digamma function are computed in closed form.

Recall that the Riemann zeta function, denoted as, is a complex function that plays a crucial role in number theory and has applications in physics, probability theory, and applied statistics. It is defined in the integral form,

(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 for sine converges to sine, namely

(Euler's product for sine).

Replacing x with ix gives

.

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 logarithm of both sides of Euler’s product for sine. By differentiating both sides and using the identities mentioned above, we obtain the following beautiful series:

,

where

is Euler's constant and its value is 0.5772156649015328606… .

Some special values

x = -1:

.

x = 1:

.

x = 1/2:

.

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

.

Special values

x = -1, it gives

.

x = 1/2, it gives

.

Digamma Function ()

The digamma function is defined as

. (II)

It is interested that differentiating (I) results in a connection to the digamma function, which satisfies the following expression:

. (III)

Special values

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

.

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

Relation Series

Series List V in relation to Euler's constant ().

References

[1*] Riemann zeta function

Riemann zeta function, http://en.wikipedia.org/wiki/Riemann_zeta_function, from Wikipedia resource.
Weisstein, Eric W. "Riemann Zeta Function." From Mathworld - A Wolfram Web Resource. http://mathworld.wolfram.com/RiemannZetaFunction.html.

[2*] Gamma function

Gamma function, http://en.wikipedia.org/wiki/Gamma_function, from Wikipedia resource.
Weisstein, Eric W. "Gamma Function." From Mathworld - A Wolfram Web Resource.

http://mathworld.wolfram.com/GammaFunction.html.

[3*] Digamma function

Digamma function, http://en.wikipedia.org/wiki/Gamma_function, from Wikipedia resource.
Weisstein, Eric W. "Digamma Function." From Mathworld - A Wolfram Web Resource.

http://mathworld.wolfram.com/DigammaFunction.html.

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

More about Riemann, especially the 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.

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

.

45.

.

46.

.

47.

.

48.

.

49.

.

50.

.

51.

.

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

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 following formulas show some irrational numbers in connection with the constant Pi.

,

where

,

which is one of the roots of the quartic equation

.

,

where

.

,

where

.

In-Text or Website Citation

Tue N. Vu, BBP-Type Series in Connection with one of the Roots of Quartic Equation, from Series Math Study Resource.

Hyperlink: http://seriesmathstudy.com/sms/irrationalnum.

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].

The first 33 values in closed-form of the Riemann zeta function at positive even integers are

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

(3/17/2009)

Even Zeta in Odd Form

References

[1] Sury, B. Bernoulli numbers and the riemann zeta function. Reson 8, 54–62 (2003). https://doi.org/10.1007/BF02834403.

In-Text or Website Citation

Tue N. Vu, The Values of Riemann Zeta Function for Positive Even Integers, from Series Math Study Resource.

Hyperlink: http://seriesmathstudy.com/sms/evenriemannzeta.

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, from Series Math Study Resource.
Hyperlink: http://www.seriesmathstudy.com/finitealternativeodd.

Power Sum

Power Sum

For each positive integer n,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

In-Text or Website Citation

Tue N. Vu, Power Sum, from Series Math Study Resource.

Hyperlink: http://seriesmathstudy.com/sms/finitepowersum.

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

Download (powersum.txt)

Other finite series:

Finite Odd Power Series

Power Sum (Text Format)

50 Identities of Power Summation

(09/26/2009 - update)

For each positive integer n,

1 + 2 + 3 + ... + n = (1/2)n² + (1/2)n .

1² + 2² + 3² + ... + n² = (1/3)n³ + (1/2)n² + (1/6)n .

1³ + 2³ + 3³ + ... + n³ = (1/4)n⁴ + (1/2)n³ + (1/4 )n² .

1⁴ + 2⁴ + 3⁴ + ... + n⁴ = (1/5)n⁵ + (1/2)n⁴ + (1/3)n³ - (1/30)n .

1⁵ + 2⁵ + 3⁵ + ... + n⁵ = (1/6)n⁶ + (1/2)n⁵ + (5/12)n⁴ - (1/12)n² .

1⁶ + 2⁶ + 3⁶ + ... + n⁶ = (1/7)n⁷ + (1/2)n⁶ + (1/2)n⁵ - (1/6)n³ + (1/42)n .

1⁷ + 2⁷ + 3⁷ + ... + n⁷ = (1/8)n⁸ + (1/2)n⁷ + (7/12)n⁶ - (7/24)n⁴ + (1/12)n² .

1⁸ + 2⁸ + 3⁸ + ... + n⁸ = (1/9)n⁹ + (1/2)n⁸ + (2/3)n⁷ - (7/15)n⁵ + (2/9)n³ - (1/30)n .

1⁹ + 2⁹ + 3⁹ + ... + n⁹ = (1/10)n¹⁰ + (1/2)n⁹ + (3/4)n⁸ - (7/10)n⁶ + (1/2)n⁴ - (3/20)n² .

1¹⁰ + 2¹⁰ + 3¹⁰ + ... + n¹⁰ = (1/11)n¹¹ + (1/2)n¹⁰ + (5/6)n⁹ - n⁷ + n⁵ - (1/2)n³ + (5/66)n .

1¹¹ + 2¹¹ + 3¹¹ + ... + n¹¹ = (1/12)n¹² + (1/2)n¹¹ + (11/12)n¹⁰ - (11/8)n⁸ + (11/6)n⁶ - (11/8)n⁴ + (5/12)n² .

1¹² + 2¹² + 3¹² + ... + n¹² = (1/13)n¹³ + (1/2)n¹² + n¹¹ - (11/6)n⁹ + (22/7)n⁷ - (33/10)n⁵ + (5/3)n³ - (691/2730)n .

1¹³ + 2¹³ + 3¹³ + ... + n¹³ = (1/14)n¹⁴ + (1/2)n¹³ + (13/12)n¹² - (143/60)n¹⁰ + (143/28)n⁸ - (143/20)n⁶ + (65/12)n⁴ - (691/420)n² .

1¹⁴ + 2¹⁴ + 3¹⁴ + ... + n¹⁴ = (1/15)n¹⁵ + (1/2)n¹⁴ + (7/6)n¹³ - (91/30)n¹¹ + (143/18)n⁹ - (143/10)n⁷ + (91/6)n⁵ - (691/90)n³ + (7/6)n .

1¹⁵ + 2¹⁵ + 3¹⁵ + ... + n¹⁵ = (1/16)n¹⁶ + (1/2)n¹⁵ + (5/4)n¹⁴ - (91/24)n¹² + (143/12)n¹⁰ - (429/16)n⁸ + (455/12)n⁶ - (691/24)n⁴ + (35/4)n² .

1¹⁶ + 2¹⁶ + 3¹⁶ + ... + n¹⁶ = (1/17)n¹⁷ + (1/2)n¹⁶ + (4/3)n¹⁵ - (14/3)n¹³ + (52/3)n¹¹ - (143/3)n⁹ + (260/3)n⁷ - (1382/15)n⁵ + (140/3)n³ - (3617/510)n .

1¹⁷ + 2¹⁷ + 3¹⁷ + ... + n¹⁷ = (1/18)n¹⁸ + (1/2)n¹⁷ + (17/12)n¹⁶ - (17/3)n¹⁴ + (221/9)n¹² - (2431/30)n¹⁰ + (1105/6)n⁸ - (11747/45)n⁶ + (595/3)n⁴ - (3617/60)n² .

1¹⁸ + 2¹⁸ + 3¹⁸ + ... + n¹⁸ = (1/19)n¹⁹ + (1/2)n¹⁸ + (3/2)n¹⁷ - (34/5)n¹⁵ + (34)n¹³ - (663/5)n¹¹ + (1105/3)n⁹ - (23494/35)n⁷ + (714)n⁵ - (3617/10)n³ + (43867/798)n .

1¹⁹ + 2¹⁹ + 3¹⁹ + ... + n¹⁹ = (1/20)n²⁰ + (1/2)n¹⁹ + (19/12)n¹⁸ - (323/40)n¹⁶ + (323/7)n¹⁴ - (4199/20)n¹² + (4199/6)n¹⁰ - (223193/140)n⁸ + (2261)n⁶ - (68723/40)n⁴ + (43867/84)n² .

1²⁰ + 2²⁰ + 3²⁰ + ... + n²⁰ = (1/21)n²¹ + (1/2)n²⁰ + (5/3)n¹⁹ - (19/2)n¹⁷ + (1292/21)n¹⁵ - (323)n¹³ + (41990/33)n¹¹ - (223193/63)n⁹ + (6460)n⁷ - (68723/10)n⁵ + (219335/63)n³ - (174611/330)n .

1²¹ + 2²¹ + 3²¹ + ... + n²¹ = (1/22)n²² + (1/2)n²¹ + (7/4)n²⁰ - (133/12)n¹⁸ + (323/4)n¹⁶ - (969/2)n¹⁴ + (146965/66)n¹² - (223193/30)n¹⁰ + (33915/2)n⁸ - (481061/20)n⁶ + (219335/12)n⁴ - (1222277/220)n² .

1²² + 2²² + 3²² + ... + n²² = (1/23)n²³ + (1/2)n²² + (11/6)n²¹ - (77/6)n¹⁹ + (209/2)n¹⁷ - (3553/5)n¹⁵ + (11305/3)n¹³ - (223193/15)n¹¹ + (124355/3)n⁹ - (755953/10)n⁷ + (482537/6)n⁵ - (1222277/30)n³ + (854513/138)n .

1²³ + 2²³ + 3²³ + ... + n²³ = (1/24)n²⁴ + (1/2)n²³ + (23/12)n²² - (1771/120)n²⁰ + (4807/36)n¹⁸ - (81719/80)n¹⁶ + (37145/6)n¹⁴ - (5133439/180)n¹² + (572033/6)n¹⁰ - (17386919/80)n⁸ + (11098351/36)n⁶ - (28112371/120)n⁴ + (854513/12)n² .

1²⁴ + 2²⁴ + 3²⁴ + ... + n²⁴ = (1/25)n²⁵ + (1/2)n²⁴ + (2)n²³ - (253/15)n²¹ + (506/3)n¹⁹ - (14421/10)n¹⁷ + (29716/3)n¹⁵ - (10266878/195)n¹³ + (208012)n¹¹ - (17386919/30)n⁹ + (22196702/21)n⁷ - (28112371/25)n⁵ + (1709026/3)n³ - (236364091/2730)n .

1²⁵ + 2²⁵ + 3²⁵ + ... + n²⁵ = (1/26)n²⁶ + (1/2)n²⁵ + (25/12)n²⁴ - (115/6)n²² + (1265/6)n²⁰ - (24035/12)n¹⁸ + (185725/12)n¹⁶ - (25667195/273)n¹⁴ + (1300075/3)n¹² - (17386919/12)n¹⁰ + (277458775/84)n⁸ - (28112371/6)n⁶ + (21362825/6)n⁴ - (1181820455/1092)n² .

1²⁶ + 2²⁶ + 3²⁶ + ... + n²⁶ = (1/27)n²⁷ + (1/2)n²⁶ + (13/6)n²⁵ - (65/3)n²³ + (16445/63)n²¹ - (16445/6)n¹⁹ + (142025/6)n¹⁷ - (10266878/63)n¹⁵ + (2600150/3)n¹³ - (20548177/6)n¹¹ + (3606964075/378)n⁹ - (52208689/3)n⁷ + (55543345/3)n⁵ - (1181820455/126)n³ + (8553103/6)n .

1²⁷ + 2²⁷ + 3²⁷ + ... + n²⁷ = (1/28)n²⁸ + (1/2)n²⁷ + (9/4)n²⁶ - (195/8)n²⁴ + (4485/14)n²² - (29601/8)n²⁰ + (142025/4)n¹⁸ - (15400317/56)n¹⁶ + (1671525)n¹⁴ - (61644531/8)n¹² + (721392815/28)n¹⁰ - (469878201/8)n⁸ + (166630035/2)n⁶ - (3545461365/56)n⁴ + (76977927/4)n² .

1²⁸ + 2²⁸ + 3²⁸ + ... + n²⁸ = (1/29)n²⁹ + (1/2)n²⁸ + (7/3)n²⁷ - (273/10)n²⁵ + (390)n²³ - (9867/2)n²¹ + (52325)n¹⁹ - (905901/2)n¹⁷ + (3120180)n¹⁵ - (33193209/2)n¹³ + (65581165)n¹¹ - (365460823/2)n⁹ + (333260070)n⁷ - (709092273/2)n⁵ + (179615163)n³ - (23749461029/870)n .

1²⁹ + 2²⁹ + 3²⁹ + ... + n²⁹ = (1/30)n³⁰ + (1/2)n²⁹ + (29/12)n²⁸ - (609/20)n²⁶ + (1885/4)n²⁴ - (26013/4)n²² + (303485/4)n²⁰ - (8757043/12)n¹⁸ + (22621305/4)n¹⁶ - (137514723/4)n¹⁴ + (1901853785/12)n¹² - (10598363867/20)n¹⁰ + (4832271015/4)n⁸ - (6854558639/4)n⁶ + (5208839727/4)n⁴ - (23749461029/60)n² .

1³⁰ + 2³⁰ + 3³⁰ + ... + n³⁰ = (1/31)n³¹ + (1/2)n³⁰ + (5/2)n²⁹ - (203/6)n²⁷ + (1131/2)n²⁵ - (16965/2)n²³ + (216775/2)n²¹ - (2304485/2)n¹⁹ + (19959975/2)n¹⁷ - (137514723/2)n¹⁵ + (731482225/2)n¹³ - (31795091601/22)n¹¹ + (8053785025/2)n⁹ - (102818379585/14)n⁷ + (15626519181/2)n⁵ - (23749461029/6)n³ + (8615841276005/14322)n .

1³¹ + 2³¹ + 3³¹ + ... + n³¹ = (1/32)n³² + (1/2)n³¹ + (31/12)n³⁰ - (899/24)n²⁸ + (2697/4)n²⁶ - (175305/16)n²⁴ + (6720025/44)n²² - (14287807/8)n²⁰ + (68751025/4)n¹⁸ - (4262956413/32)n¹⁶ + (22675948975/28)n¹⁴ - (328549279877/88)n¹² + (49933467155/4)n¹⁰ - (3187369767135/112)n⁸ + (161474031537/4)n⁶ - (736233291899/24)n⁴ + (8615841276005/924)n² .

1³² + 2³² + 3³² + ... + n³² = (1/33)n³³ + (1/2)n³² + (8/3)n³¹ - (124/3)n²⁹ + (7192/9)n²⁷ - (70122/5)n²⁵ + (2337400/11)n²³ - (57151228/21)n²¹ + (28947800)n¹⁹ - (4262956413/17)n¹⁷ + (36281518360/21)n¹⁵ - (101092086116/11)n¹³ + (36315248840)n¹¹ - (2124913178090/21)n⁹ + (184541750328)n⁷ - (2944933167596/15)n⁵ + (68926730208040/693)n³ - (7709321041217/510)n .

1³³ + 2³³ + 3³³ + ... + n³³ = (1/34)n³⁴ + (1/2)n³³ + (11/4)n³² - (682/15)n³⁰ + (19778/21)n²⁸ - (89001/5)n²⁶ + (292175)n²⁴ - (28575614/7)n²² + (47763870)n²⁰ - (15630840181/34)n¹⁸ + (49887087745/14)n¹⁶ - (21662589882)n¹⁴ + (99866934310)n¹² - (2337404495899/7)n¹⁰ + (761234720103)n⁸ - (16197132421778/15)n⁶ + (17231682552010/21)n⁴ - (84802531453387/340)n² .

1³⁴ + 2³⁴ + 3³⁴ + ... + n³⁴ = (1/35)n³⁵ + (1/2)n³⁴ + (17/6)n³³ - (748/15)n³¹ + (23188/21)n²⁹ - (336226/15)n²⁷ + (397358)n²⁵ - (42242212/7)n²³ + (77331980)n²¹ - (822675799)n¹⁹ + (49887087745/7)n¹⁷ - (245509351996/5)n¹⁵ + (261190443580)n¹³ - (7224704805506/7)n¹¹ + (2875775609278)n⁹ - (78671786048636/15)n⁷ + (117175441353668/21)n⁵ - (84802531453387/30)n³ + (2577687858367/6)n .

1³⁵ + 2³⁵ + 3³⁵ + ... + n³⁵ = (1/36)n³⁶ + (1/2)n³⁵ + (35/12)n³⁴ - (1309/24)n³² + (11594/9)n³⁰ - (168113/6)n²⁸ + (534905)n²⁶ - (52802765/6)n²⁴ + (123028150)n²² - (5758730593/4)n²⁰ + (249435438725/18)n¹⁸ - (429641365993/4)n¹⁶ + (652976108950)n¹⁴ - (18061762013765/6)n¹² + (10065214632473)n¹⁰ - (137675625585113/6)n⁸ + (292938603384170/9)n⁶ - (593617720173709/24)n⁴ + (90219075042845/12)n² .

1³⁶ + 2³⁶ + 3³⁶ + ... + n³⁶ = (1/37)n³⁷ + (1/2)n³⁶ + (3)n³⁵ - (119/2)n³³ + (1496)n³¹ - (34782)n²⁹ + (2139620/3)n²⁷ - (63363318/5)n²⁵ + (192565800)n²³ - (2468027397)n²¹ + (498870877450/19)n¹⁹ - (227457193761)n¹⁷ + (1567142661480)n¹⁵ - (108370572082590/13)n¹³ + (32940702433548)n¹¹ - (275351251170226/3)n⁹ + (1171754413536680/7)n⁷ - (1780853160521127/10)n⁵ + (90219075042845)n³ - (26315271553053477373/1919190)n .

1³⁷ + 2³⁷ + 3³⁷ + ... + n³⁷ = (1/38)n³⁸ + (1/2)n³⁷ + (37/12)n³⁶ - (259/4)n³⁴ + (6919/4)n³² - (214489/5)n³⁰ + (2827355/3)n²⁸ - (1172221383/65)n²⁶ + (296872275)n²⁴ - (8301546699/2)n²² + (1845822246565/38)n²⁰ - (935101796573/2)n¹⁸ + (7248034809345/2)n¹⁶ - (2004855583527915/91)n¹⁴ + (101567165836773)n¹² - (5093998146649181/15)n¹⁰ + (5419364162607145/7)n⁸ - (21963855646427233/20)n⁶ + (3338105776585265/4)n⁴ - (26315271553053477373/103740)n² .

1³⁸ + 2³⁸ + 3³⁸ + ... + n³⁸ = (1/39)n³⁹ + (1/2)n³⁸ + (19/6)n³⁷ - (703/10)n³⁵ + (11951/6)n³³ - (262922/5)n³¹ + (3704810/3)n²⁹ - (14848137518/585)n²⁷ + (451245858)n²⁵ - (6857799447)n²³ + (1845822246565/21)n²¹ - (935101796573)n¹⁹ + (8100744786915)n¹⁷ - (5078967478270718/91)n¹⁵ + (296888638599798)n¹³ - (17597448142969898/15)n¹¹ + (205935838179071510/63)n⁹ - (59616179611731061/10)n⁷ + (12684801951024007/2)n⁵ - (26315271553053477373/8190)n³ + (2929993913841559/6)n .

1³⁹ + 2³⁹ + 3³⁹ + ... + n³⁹ = (1/40)n⁴⁰ + (1/2)n³⁹ + (13/4)n³⁸ - (9139/120)n³⁶ + (9139/4)n³⁴ - (5126979/80)n³² + (4816253/3)n³⁰ - (7424068759/210)n²⁸ + (676868787)n²⁶ - (89151392811/8)n²⁴ + (2181426291395/14)n²² - (36468970066347/20)n²⁰ + (35103227409965/2)n¹⁸ - (7618451217406077/56)n¹⁶ + (827046921813723)n¹⁴ - (114383412929304337/30)n¹² + (267716589632792963/21)n¹⁰ - (2325031004857511379/80)n⁸ + (164902425363312091/4)n⁶ - (26315271553053477373/840)n⁴ + (38089920879940267/4)n² .

1⁴⁰ + 2⁴⁰ + 3⁴⁰ + ... + n⁴⁰ = (1/41)n⁴¹ + (1/2)n⁴⁰ + (10/3)n³⁹ - (247/3)n³⁷ + (18278/7)n³⁵ - (155363/2)n³³ + (6214520/3)n³¹ - (1024009484/21)n²⁹ + (3008305720/3)n²⁷ - (89151392811/5)n²⁵ + (1896892427300/7)n²³ - (3473235244414)n²¹ + (36950765694700)n¹⁹ - (2240720946295905/7)n¹⁷ + (2205458458169928)n¹⁵ - (35194896285939796/3)n¹³ + (10708663585311718520/231)n¹¹ - (775010334952503793/6)n⁹ + (235574893376160130)n⁷ - (26315271553053477373/105)n⁵ + (380899208799402670/3)n³ - (261082718496449122051/13530)n .

1⁴¹ + 2⁴¹ + 3⁴¹ + ... + n⁴¹ = (1/42)n⁴² + (1/2)n⁴¹ + (41/12)n⁴⁰ - (533/6)n³⁸ + (374699/126)n³⁶ - (374699/4)n³⁴ + (31849415/12)n³² - (20992194422/315)n³⁰ + (4405019090/3)n²⁸ - (281169777327/10)n²⁶ + (19443147379825/42)n²⁴ - (71201322510487/11)n²² + (75749069674135)n²⁰ - (30623186266044035/42)n¹⁸ + (11302974598120881/2)n¹⁶ - (103070767694537974/3)n¹⁴ + (109763801749445114830/693)n¹² - (31775423733052655513/60)n¹⁰ + (4829285314211282665/4)n⁸ - (1078926133675192572293/630)n⁶ + (7808433780387754735/6)n⁴ - (261082718496449122051/660)n² .

1⁴² + 2⁴² + 3⁴² + ... + n⁴² = (1/43)n⁴³ + (1/2)n⁴² + (7/2)n⁴¹ - (287/3)n³⁹ + (10127/3)n³⁷ - (1124097/10)n³⁵ + (222945905/66)n³³ - (1354335124/15)n³¹ + (2126560940)n²⁹ - (656062813763/15)n²⁷ + (777725895193)n²⁵ - (130019806323498/11)n²³ + (151498139348270)n²¹ - (1611746645581265)n¹⁹ + (13962498032972853)n¹⁷ - (1442990747723531636/15)n¹⁵ + (16886738730683863820/33)n¹³ - (20220724193760780781/10)n¹¹ + (33804997199478978655/6)n⁹ - (1078926133675192572293/105)n⁷ + (10931807292542856629)n⁵ - (1827579029475143854357/330)n³ + (1520097643918070802691/1806)n .

1⁴³ + 2⁴³ + 3⁴³ + ... + n⁴³ = (1/44)n⁴⁴ + (1/2)n⁴³ + (43/12)n⁴² - (12341/120)n⁴⁰ + (22919/6)n³⁸ - (16112057/120)n³⁶ + (563921995/132)n³⁴ - (14559102583/120)n³² + (9144212042/3)n³⁰ - (4030100141687/60)n²⁸ + (2572477961023/2)n²⁶ - (931808611985069/44)n²⁴ + (296109999635255)n²² - (13861021151998879/4)n²⁰ + (66709712824203631/2)n¹⁸ - (15512150538027965087/60)n¹⁶ + (363064882709703072130/231)n¹⁴ - (869491140331713573583/120)n¹² + (290722975915519216433/12)n¹⁰ - (46393823748033280608599/840)n⁸ + (470067713579342835047/6)n⁶ - (78585898267431185737351/1320)n⁴ + (1520097643918070802691/84)n² .

1⁴⁴ + 2⁴⁴ + 3⁴⁴ + ... + n⁴⁴ = (1/45)n⁴⁵ + (1/2)n⁴⁴ + (11/3)n⁴³ - (3311/30)n⁴¹ + (38786/9)n³⁹ - (4790071/30)n³⁷ + (16112057/3)n³⁵ - (14559102583/90)n³³ + (12978881608/3)n³¹ - (1528658674433/15)n²⁹ + (56594515142506/27)n²⁷ - (931808611985069/25)n²⁵ + (13028839983951220/23)n²³ - (152471232671987669/21)n²¹ + (77242825375393678)n¹⁹ - (10037273877547506821/15)n¹⁷ + (290451906167762457704/63)n¹⁵ - (735723272588373023801/30)n¹³ + (290722975915519216433/3)n¹¹ - (510332061228366086694589/1890)n⁹ + (1477355671249363195862/3)n⁷ - (78585898267431185737351/150)n⁵ + (16721074083098778829601/63)n³ - (27833269579301024235023/690)n .

1⁴⁵ + 2⁴⁵ + 3⁴⁵ + ... + n⁴⁵ = (1/46)n⁴⁶ + (1/2)n⁴⁵ + (15/4)n⁴⁴ - (473/4)n⁴² + (19393/4)n⁴⁰ - (756327/4)n³⁸ + (80560285/12)n³⁶ - (856417799/4)n³⁴ + (24335403015/4)n³² - (1528658674433/10)n³⁰ + (141486287856265/42)n²⁸ - (645098269835817/10)n²⁶ + (48858149939817075/46)n²⁴ - (207915317279983185/14)n²² + (347592714189271551/2)n²⁰ - (10037273877547506821/6)n¹⁸ + (181532441354851536065/14)n¹⁶ - (315309973966445581629/4)n¹⁴ + (1453614879577596082165/4)n¹² - (510332061228366086694589/420)n¹⁰ + (11080167534370223968965/4)n⁸ - (78585898267431185737351/20)n⁶ + (83605370415493894148005/28)n⁴ - (83499808737903072705069/92)n² .

1⁴⁶ + 2⁴⁶ + 3⁴⁶ + ... + n⁴⁶ = (1/47)n⁴⁷ + (1/2)n⁴⁶ + (23/6)n⁴⁵ - (253/2)n⁴³ + (10879/2)n⁴¹ - (446039/2)n³⁹ + (50078015/6)n³⁷ - (19697609377/70)n³⁵ + (16961038465/2)n³³ - (1134166113289/5)n³¹ + (112213262782555/21)n²⁹ - (1648584467358199/15)n²⁷ + (1954325997592683)n²⁵ - (207915317279983185/7)n²³ + (380696782207297413)n²¹ - (12150384167557508257/3)n¹⁹ + (245602714774210901735/7)n¹⁷ - (2417376467076082792489/10)n¹⁵ + (2571780171560362299215/2)n¹³ - (1067057946204765453997777/210)n¹¹ + (84947951096838383762065/6)n⁹ - (258210808592988181708439/10)n⁷ + (384584703911271913080823/14)n⁵ - (27833269579301024235023/2)n³ + (596451111593912163277961/282)n .

1⁴⁷ + 2⁴⁷ + 3⁴⁷ + ... + n⁴⁷ = (1/48)n⁴⁸ + (1/2)n⁴⁷ + (47/12)n⁴⁶ - (1081/8)n⁴⁴ + (511313/84)n⁴² - (20963833/80)n⁴⁰ + (123877195/12)n³⁸ - (925787640719/2520)n³⁶ + (46892282815/4)n³⁴ - (53305807324583/160)n³² + (1054804670156017/126)n³⁰ - (11069067137976479/60)n²⁸ + (7065640145142777/2)n²⁶ - (3257339970719736565/56)n²⁴ + (1626613523976634401/2)n²² - (571068055875202888079/60)n²⁰ + (11543327594387912381545/126)n¹⁸ - (113616693952575891246983/160)n¹⁶ + (17267666866191004009015/4)n¹⁴ - (50151723471623976337895519/2520)n¹² + (798510740310280807363411/12)n¹⁰ - (12135908003870444540296633/80)n⁸ + (18075481083829779914798681/84)n⁶ - (1308163670227148139046081/8)n⁴ + (596451111593912163277961/12)n² .

1⁴⁸ + 2⁴⁸ + 3⁴⁸ + ... + n⁴⁸ = (1/49)n⁴⁹ + (1/2)n⁴⁸ + (4)n⁴⁷ - (2162/15)n⁴⁵ + (47564/7)n⁴³ - (1533939/5)n⁴¹ + (38116060/3)n³⁹ - (50042575174/105)n³⁷ + (16077354108)n³⁵ - (4845982484053/10)n³³ + (272207656814456/21)n³¹ - (1526767881100204/5)n²⁹ + (18841707053714072/3)n²⁷ - (3908807964863683878/35)n²⁵ + (1697335851106053288)n²³ - (326324603357258793188/15)n²¹ + (4860348460794910476440/21)n¹⁹ - (340850081857727673740949/170)n¹⁷ + (13814133492952803207212)n¹⁵ - (100303446943247952675791038/1365)n¹³ + (290367541931011202677604)n¹¹ - (12135908003870444540296633/15)n⁹ + (72301924335319119659194724/49)n⁷ - (7848982021362888834276486/5)n⁵ + (2385804446375648653111844/3)n³ - (5609403368997817686249127547/46410)n .

1⁴⁹ + 2⁴⁹ + 3⁴⁹ + ... + n⁴⁹ = (1/50)n⁵⁰ + (1/2)n⁴⁹ + (49/12)n⁴⁸ - (2303/15)n⁴⁶ + (7567)n⁴⁴ - (3579191/10)n⁴² + (93384347/6)n⁴⁰ - (9218369111/15)n³⁸ + (65649195941/3)n³⁶ - (237453141718597/340)n³⁴ + (238181699712649/12)n³² - (37405813086954998/75)n³⁰ + (32972987343999626/3)n²⁸ - (13680827877022893573/65)n²⁶ + (3465394029341525463)n²⁴ - (726813889295712766646/15)n²² + (1701121961278218666754/3)n²⁰ - (5567218003676218671102167/1020)n¹⁸ + (169223135288671839288347/4)n¹⁶ - (50151723471623976337895519/195)n¹⁴ + (3557002388654887232800649/3)n¹² - (594659492189651782474535017/150)n¹⁰ + (18075481083829779914798681/2)n⁸ - (64100019841130258813257969/5)n⁶ + (29226104468101696000620089/3)n⁴ - (39265823582984723803743892829/13260)n² .

1⁵⁰ + 2⁵⁰ + 3⁵⁰ + ... + n⁵⁰ = (1/51)n⁵¹ + (1/2)n⁵⁰ + (25/6)n⁴⁹ - (490/3)n⁴⁷ + (75670/9)n⁴⁵ - (416185)n⁴³ + (56941675/3)n⁴¹ - (92183691110/117)n³⁹ + (88715129650/3)n³⁷ - (33921877388371/34)n³⁵ + (541322044801475/18)n³³ - (2413278263674516/3)n³¹ + (56849978179309700/3)n²⁹ - (45602759590076311910/117)n²⁷ + (6930788058683050926)n²⁵ - (316006038824222942020/3)n²³ + (12150871151987276191100/9)n²¹ - (1465057369388478597658465/102)n¹⁹ + (248857551895105646012275/2)n¹⁷ - (100303446943247952675791038/117)n¹⁵ + (13680778417903412433848650/3)n¹³ - (594659492189651782474535017/33)n¹¹ + (451887027095744497869967025/9)n⁹ - (91571456915900369733225670)n⁷ + (292261044681016960006200890/3)n⁵ - (196329117914923619018719464145/3978)n³ + (495057205241079648212477525/66)n.

Download (powersum.txt)

View Series in Summation Notation (17 identities)

(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)

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

November 11, 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.

Table - Computation of Exact and Approximation Harmonic Numbers

n	(Exact)	(Approximation)
1	1	1.0000364756158384
2	1.5	1.500001060257485
3	1.8333333333333333	1.8333334197475766
4	2.083333333333333	2.0833333459100944
5	2.283333333333333	2.2833333359731323
6	2.45	2.4500000007120852
7	2.5928571428571425	2.5928571430876115
8	2.7178571428571425	2.7178571429427802
9	2.8289682539682537	2.828968254003711
10	2.9289682539682538	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

In-Text or Website Citation

Tue N. Vu, A New Approximation Formula for Computing the N-th Harmonic Number (Update), from Series Math Study Resource.

Hyperlink: http://www.seriesmathstudy.com/sms/ApproximateFormulaHarmonicNum.

A Note on Evaluating the Integral of Cos(1/4 arccos(x))

(10/28/2023)

Evaluate

where n is real number.

Solution

Let

Hence, we get

Substituting these into the given integral, we get

For n = 4, we obtain

Notes:

1. The Wolfram integral calculator provided a lengthy result for the integral of cos(1/4 arccos(x)) on August 28, 2023. See the results in Figure 1. Readers can also verify the results on the old versions of the Wolfram’s calculator early in 2024.

Figure 1

2. When readers/users encounter a particularly difficult problem (e.g., solving an equation, evaluating an integral), Wolfram’s online calculator may return a numeric result with a note stating,

Figure 2

This message often appears when the problem is unsolvable, giving the impression that Wolfram can solve it. To verify this, readers can input any unsolvable integral into the Wolfram calculator.

We have two points to discuss regarding this scenario:

a. Wolfram’s calculator may not have enough time to respond to a difficult problem, prompting the message, “Try again with Pro computation time.”

b. Wolfram’s calculator may not be able to provide a closed-form solution, returning only a numeric result. In such cases, it would be more transparent and respectful for Wolfram to inform readers/users, for example, “The solution cannot be solved at this time.” This approach would give credit to students or others who might later provide a solution. Eventually, Wolfram could learn the solution from these users or from published results. Mathematicians sometimes recognize a closed-form solution without detailed steps, and they can solve it. Wolfram should respect readers by acknowledging that not all problems are currently solvable, rather than implying they can solve any problem.

Note:

If readers try to check this integral, cos(1/4 arccos(x)), on Wolfram’s calculator today, it shows the "definite integral" with the correct result for a few seconds (in a flash) and then disappears. The solution for 'definite integral' of cos(1/4 arccos(x)) was previously unseen. This behavior appears similarly in other difficult integrals, which are understandably challenging at first. After giving an overall review, a friend jokes that it's a sneaky form of intellectual theft, even though it's not worth a dime at that moment. However, for Wolfram marketing, it's like accumulating a pocketbook of pennies, slowly but surely adding up.

(Updated on 10/8/2024)

Tracey Vu's Comments

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)n⁵ - (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 + 2464n⁸- (10912/3)n⁶+ 2794n⁴ - (2555/3)n² .

12. 1¹² + 3¹² + 5¹² + ... + (2n-1)¹² = (4096/13)n¹³ - (2048)n¹¹ + (19712/3)n⁹- (87296/7)n⁷+ (67056/5)n⁵- (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.

Note that

.

Series List I

June 25, 2005

or

or

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 such that x ≠ -k and x ≠ -(k+1).

.

Special values

When n tends to infinity, (I) is reduced to a simple form

.

Let x = 0,

.

Similarly, (II) gives

. (Correct - 11/17/2009)

At x = 1,

.

Example

It is easy to verify that for n = 2, both sides of (I) are equal for all x ≠ - 1, -2, and -3, namely

.

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, from Series Math Study Resource.

Hyperlink: http://seriesmathstudy.com/sms/nov_07_2009.

Finite Series in Connection with Apéry, Pi Constant

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

(I)

(II)

The notationsandin (I) and (II) represent the special values of a new generic formula, which we define 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 expressed as

.

Special Case

n = ∞:

As n approaches to infinity, both series (I) and (II) converge to the following values:

,

where is Apéry's constant.

.

Example

If we put n = 2, both sides of (II) are equal to 29/31752, namely

.

(November 26, 2009 - Happy Thanksgiving)

Question:

For any positive integers m₁, m₂and 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, 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, 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.

n = 1

.

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

.

n = 2

.

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

,

you will obtain this series,

.

n = 3

.

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, from Series Math Study Resource.

Hyperlink: http://www.seriesmathstudy.com/sms/jan17_2010_link.

Series List II

July 5, 2005

1.

2.

3.

4.

5.

6.

7.

8.

9.

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

Series List IV

(November 12, 2005)

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Series List V

(December 19, 2005)

,

,

,

,

,

,

,

,

,

,

,

,

,

where is the Euler constant.

In-Text or Website Citation

Tue N. Vu, Series List V, 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

Return SMS

Number is the within of all thing. (Pythagoras)

Pi

(Pi)

The constant pi, denoted as

, is a real number that appears throughout nature. It is known to be both irrational and transcendental, meaning its exact value cannot be expressed as a simple fraction, and it is not a solution to any non-constant polynomial equation with rational coefficients. The value of

has been calculated with increasing precision over the centuries, and its decimal representation is infinite and non-repeating. Today,

is known to over one trillion decimal places, thanks to methods such as the Gregory-Leibniz series, Maclaurin series, and Machin-like formulas. More recently, the BBP-like series has been developed, allowing the computation of the n^thbinary or hexadecimal digit of

without needing to calculate the preceding (n-1) digits. Below are examples of BBP-like series with bases of 16 and 4096.

1. The formulas for the BBP-like series with the base of 16 are

.

(This BBP-like series was discovered by Bailey-Borwein-Plouffe in 1995.)

.

.

2. The formulas, BBP-like series, have been discovered in the base of 4096, cause the infinite series to converge extraordinarily rapidly, namely

.

(posted May 29, 2006)

.

(posted July 4, 2006)

(Click here to view in full window.)

3. Other infinite series, which yield results connecting to pi, are shown below:

.

or

.

(June 20, 2005)

.

.

(05/29/2006)

(07/04/2006 - update)

(03/17/2009 - update)

More Related Series

Other BBP-type series.
BBP-type series in Connection to one of the Roots of Quartic Equation.
BBP-type series in connection ln2 and ln3.

Main SMS

In-Text or Website Citation

Tue N. Vu, Pi, from Series Math Study Resource.

Hyperlink: http://seriesmathstudy.com/sms/pi1.

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 examine certain series related to the harmonic series and establish expressions involving recurrence relations to harmonic numbers.

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 has been discovered and has the potential to compute the constant ln(2) to billions of decimal places.

.

Notice that the base of above infinite series is

,

(August 1, 2006)

Click here to view this formula in full window size.
Series for computing the constant ln(3).

In-Text or Website Citation

Tue N. Vu, A New Formula For Computing Constant ln(2), from Series Math Study Resource.
Hyperlink: http://www.seriesmathstudy.com/log2.

Constant ln(3)

The New Formula For Computing Constant ln(3)

The following new infinite series has been discovered and has the potential to compute the constant ln(3) to billions of decimal places.

Notice:

, where k=0,1,2,3,…

(August 6, 2007)

Click here to view this formula in full window size.
Series for computing the constant ln(2).

In-Text or Website Citation

Tue N. Vu, A New Formula For Computing Constant ln(3), 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.

Machin-Type Formula

Machin-Type Formulas

(April 09, 2009)

Discovering a Family of Machin-Type Formulas through Finite Series

Entry 1.

For each positive integer n,
.

Formula (I) demonstrates that for each value of n, a Machin-type formula is obtained. As n increases, an infinite number of formulas with this structure emerge, forming a family of Machin-type formulas, as shown 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. This 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 for n = 0, 1, 2, 3, ..., infinity.

Collection of Machin-type 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, from Series Math Study Resource.

Hyperlink: http://seriesmathstudy.com/sms/machintypetv.

Other Series

Other Series

Define

,

where i, k and n are positive integers.

Setting

,

Then, as n approaches infinity, we have

or

.

The above series is expressed in the summation notation as

.

Multiplying both sides of above sum by

gives

.

Therefore, we obtain

.

We thus obtain a remarkable formula that shows the connection to Euler Constant, sums of power, and Zeta function.

,

where

.

Notice that

, 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.

A family of Power Sum formulas are formed for p = 1, 2, ..., 14 as shown in the following:

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 gives

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

(May 1, 2008)

Similar series - Finite Alternative Odd Power Series.

In-Text or Website Citation

Tue N. Vu, Power Sum and Sum of Partial Power Sums, from Series Math Study Resource.

Hyperlink: http://seriesmathstudy.com/sms/PowerSumandSumofPowerSum.

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.

.

.

.

.

Note that k starts at 2. As n tends to infinity, it is easy to see this series converges to 1/4,

.

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

.

. (correct)

.

.

.

.

.

.

The last 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, the series can be re-indexed, namely