 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 from and are 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 .

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 , and so 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 , and by 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 of or from On-Line Encyclopedia of Integer Sequences site.  Click here to quickly view it.  We also found the recurrence formula of is 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.

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)