The first Smarandache constant is defined as

where is the Smarandache Function. Cojocaru and Cojocaru (1996a) prove that exists and is bounded by . The lower limit given above is obtained by taking 40,000 terms of the sum.

Cojocaru and Cojocaru (1996b) prove that the second Smarandache constant

is an Irrational Number.

Cojocaru and Cojocaru (1996c) prove that the series

converges to a number , and that

converges for a fixed Real Number . The values for small are

Sandor (1997) shows that the series

converges to an Irrational. Burton (1995) and Dumitrescu and Seleacu (1996) show that the series

converges. Dumitrescu and Seleacu (1996) show that the series

and

converge for a natural number (which must be nonzero in the latter case). Dumitrescu and Seleacu (1996) show that

converges. Burton (1995) and Dumitrescu and Seleacu (1996) show that the series

and

converge for .

**References**

Burton, E. ``On Some Series Involving the Smarandache Function.'' *Smarandache Notions J.* **6**, 13-15, 1995.

Burton, E. ``On Some Convergent Series.'' *Smarandache Notions J.* **7**, 7-9, 1996.

Cojocaru, I. and Cojocaru, S. ``The First Constant of Smarandache.'' *Smarandache Notions J.* **7**, 116-118, 1996a.

Cojocaru, I. and Cojocaru, S. ``The Second Constant of Smarandache.'' *Smarandache Notions J.* **7**, 119-120, 1996b.

Cojocaru, I. and Cojocaru, S. ``The Third and Fourth Constants of Smarandache.'' *Smarandache Notions J.* **7**, 121-126, 1996c.

``Constants Involving the Smarandache Function.'' http://www.gallup.unm.edu/~smarandache/CONSTANT.TXT.

Dumitrescu, C. and Seleacu, V. ``Numerical Series Involving the Function .'' *The Smarandache Function.*
Vail: Erhus University Press, pp. 48-61, 1996.

Ibstedt, H. *Surfing on the Ocean of Numbers--A Few Smarandache Notions and Similar Topics.*
Lupton, AZ: Erhus University Press, pp. 27-30, 1997.

Sandor, J. `On The Irrationality Of Certain Alternative Smarandache Series.'' *Smarandache Notions J.* **8**, 143-144, 1997.

Smarandache, F. *Collected Papers, Vol. 1.* Bucharest, Romania: Tempus, 1996.

Smarandache, F. *Collected Papers, Vol. 2.* Kishinev, Moldova: Kishinev University Press, 1997.

© 1996-9

1999-05-26