Collection of Well Known Series

Collection of Well Known Series

 

Leonhard Euler

Some Special Leonhard Euler’s Series and Formulas
(1707-1783)
 
 
 
  • Every even integer ( ) is the sum of two odd primes.

 

  • Every odd integer ( ) is the sum of three odd primes.

 

 

 

 

 

 

 

 

 

 

 

 

  • (definition).

 

 

 

 

  • (Euler Sum) (Mathematical Montly, Euler also considered this sum in response to a letter from Goldbach.)

 Series Collections

 

Quick View

Quick view of some series and formulas.

Srinivasa Aiyangar Ramanujan

Some Special Srinivasa Aiyangar Ramanujan's Series and Formulas
(1887-1920)
 
 
 
  •     ,

 

where and for each positive integer k,
 
 
  •  

 

 
  • .

 

 
  •  .

 

 
  •  .

 

 
  • .

 

 

  • .

 

 
  • .

 

 
  • .

 

 
  • .

 

 
  • .

 

 
  • , for all , where is the Gamma Function.
 
 
  • Define , which is called Ramanujan Theta Function. Special values include

 

 
.
 
 
  • Hypergeometric Identity

 

.
 
 
  • Ramanujan’s Master Theorem

 

then
 
for small of x = 0.
 
 
 
Srinivasa Aiyangar Ramanujan is the man who masterminded modular forms, limits, and the law for recursive logics. (T.V.)

 Series Collections

 
References
Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, 1985.
Berndt, B. C. Ramanujan's Notebooks: Part II. New York: Springer-Verlag, 1989.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.
Berndt, B. C. and Bhargava, S. ``A Remarkable Identity Found in Ramanujan's Third Notebook.'' Glasgow Math. J, 1992.
D.V. Chudnovsky and G.V. Chudnovsky. Approxiamtions and Complex Multiplication According to Ramanujan. San Diego, CA: Academic Press, 1988.

Well Known Series

Well Known Series
 
 
Constants
 
1.                                             (Wallis)
 
2.       (Vieta)
 
3.                                (Barnes)
 
4.                                        (Barnes)
 
5.                                            (Demys)
 
6.                                     (Glaisher)
 
7.       (Flajolet –Vardi)
 
8.                            (Roger Apery)
 
9.            (Leibniz-Gregory)
 
10.                       (Newton)
 
11.       (Knopp)
 
12.                               (Flajolet-Vardi)
 
13.                                 (Knopp)
 
14.                                   (Sharp)
 
15.                                                     (Euler)             
 
16.                                (Euler)
 
17.                                            (Euler)
 
18.                               (Ramanujan)
 
19.                          (Lucas)
 
20.               (BBP)
 
Series Collections
If I were again beginning my studies, I would follow the advice of Plato and start with mathematics.
 Galileo Galilei