Series Math Study

God made the natural numbers; all the rest is the work of man.      - Leopold Kronecker

The purpose of Series Math Study (SMS) website is to explore and develop the beauty of series formulas that have connections to special mathematical constants.  Most of the series in this website are found with completed proofs.  Some series can be used to calculate ever more accurate values of some special math constants as the desire that has challenged mathematicians for many centuries.

I am ashamed to tell you to how many figures I carried these computations, having no other business at the time.

Issac Newton, personal journal, 1666

(May 1, 2008)

Power Sum and Sum of Partial Power Sum. Click here to see other forms.

(Jan 31, 2008)

(May 15, 2007)

The general series formula below is true for all |x|a and a1, namely

.

A special case as a = 1 and x = 0, it gives

.

Click here to see other form.

(April 07, 2007)

The fast conergent series are used for computing the logarithm constants log 2 and log 3 (updated).

(February 14, 2007)

.

(Noticeare the special values of the Riemann zeta function at positive integers)

(December 23, 2006)

The formula below is true for all .

.

Click here to see more formulas and examples.

(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)

The sums of the following identities are true for any positive integers n.

1.

2.

3.

4.

More formulas and examples of this type are found in this link.  We do not know whether the type of these formulas is new.

(September 17, 2006)

1.

2.

(August 25, 2006)

GraphFunc is an online program for drawing graphs of basic mathematical functions in 2D and 3D coordinate systems.  Click here to use this tool.

(August 1, 2006)

1.

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

(July 4, 2006)

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

(May 29, 2006)

A fast convergent series of BPP-type formulas has been found and can be used for computing the n-th digit ofin base 4096 without computing any prior digits. 

.

(We do not know whether this series is new)

(May 28, 2006)

1. 

2. 

(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)

1. 

2. 

(April 21, 2006)

(March 7, 2006)

or it can be written in product form:

.

(March 6, 2006)

or it can be written in product form:, which is

.

(March 4, 2006)

, where is the Euler constant.

(January 16, 2006)

1.  (new?),  whereis Apéry’s constant.

2.  (new?)

3. (new?)

4. (new?)

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

6. 

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