CONSTANTS INVOLVING THE SMARANDACHE FUNCTION
                           
           


Let S(n) be the Smarandache Function, i.e. the smallest integer such that
S(n)! is divisible by n.

1)  The First Constant of Smarandache:

      Sigma(1/S(n)!) is convergent to a number s1 between 0.000 and 0.717.
       n>=2

Reference:
  [1] I.Cojocaru, S.Cojocaru, "The First Constant of Smarandache", in
      <Smarandache Notions Journal>, University of Craiova, Vol. 7, 
      No. 1-2-3, pp. 116-118, August 1996.




2)  The Second Constant of Smarandache:
      
      Sigma(S(n)/n!) is convergent to an irrational number s2.
       n>=2

Reference:
  [1] I.Cojocaru, S.Cojocaru, "The Second Constant of Smarandache", in
      <Smarandache Notions Journal>, University of Craiova, Vol. 7, 
      No. 1-2-3, pp. 119-120, August 1996.




3)  The Third Constant of Smarandache:

      Sigma(1/(S(2)S(3)...S(n))) is convergent to a number s3, which is 
       n>=2

    between 0.71 and 1.01.

Reference:
  [1] I.Cojocaru, S.Cojocaru, "The Third and Fourth Constants of Smarandache",
      in <Smarandache Notions Journal>, University of Craiova, Vol. 7, No.
      1-2-3, pp. 121-126, August 1996.



4)  The Fourth Constant of Smarandache:

      Sigma(n^alpha/(S(2)S(3)...S(n))), where alpha >= 1,
       n>=2

      is convergent to a number s4.

Reference:
  [1] I.Cojocaru, S.Cojocaru, "The Third and Fourth Constants of Smarandache",
      in <Smarandache Notions Journal>, University of Craiova, Vol. 7, No.
      1-2-3, pp. 121-126, August 1996.


5)  The series

                       n-1
             Sigma (-1)   (S(n)/n!)
             n>=1

   converges to an irrational number.

Reference:
 [1] Sandor, Jozsef, "On The Irrationality Of Certain Alternative Smarandache
     Series", <Smarandache Notions Journal>, Vol. 8, No. 1-2-3, Fall 1997,
     pp. 143-144.




6)  The series

                       S(n)
                Sigma --------
                n>=2   (n+1)!

    converges to a number s6, where e-3/2 < s6 < 1/2.


Reference:
 [1] Burton, Emil, "On Some Series Involving the Smarandache Function",
     <Smarandache Function Journal>, Vol. 6, No. 1, June 1995, ISSN 1053-4792,
     pp. 13-15.
 [2] Dumitrescu, C., Seleacu, V., "The Smarandache Function", Erhus University
     Press, Vail, USA, 1996, pp. 48-61 (chapter "Numerical Series Containing
     the Function S").



7)  The series

                       S(n)
                Sigma --------, where r is a natural number,
                n>=r   (n+r)!

    converges to a number s7.


Reference:
 [1] Dumitrescu, C., Seleacu, V., "The Smarandache Function", Erhus University
     Press, Vail, USA, 1996, pp. 48-61 (chapter "Numerical Series Containing
     the Function S").



8)  The series

                       S(n)
                Sigma --------, where r is a nonzero natural number,
                n>=r   (n-r)!

    converges to a number s8.


Reference:
 [1] Dumitrescu, C., Seleacu, V., "The Smarandache Function", Erhus University
     Press, Vail, USA, 1996, pp. 48-61 (chapter "Numerical Series Containing
     the Function S").



9)  The series

                            1
              Sigma --------------------
              n>=2     n      
                       Sigma (S(i)!/i)
                       i=2


   is convergent to a number s9.


Reference:
 [1] Dumitrescu, C., Seleacu, V., "The Smarandache Function", Erhus University
     Press, Vail, USA, 1996, pp. 48-61 (chapter "Numerical Series Containing
     the Function S").



10) The series 

                               1
                Sigma -------------------, where alpha > 1,
                n>=2       alpha   ______
                       S(n)      \/S(n)!


    is convergent to a number s10.



References:
 [1] Burton, Emil, "On Some Convergent Series", <Smarandache Notions Journal>,
     Vol. 7, No. 1-2-3, August 1996, pp. 7-9.
 [2] Dumitrescu, C., Seleacu, V., "The Smarandache Function", Erhus University
     Press, Vail, USA, 1996, pp. 48-61 (chapter "Numerical Series Containing
     the Function S").




11) The series 

                               1
                Sigma -----------------------, where alpha > 1,
                n>=2       alpha   _________
                       S(n)      \/(S(n)-1)!


    is convergent to a number s11.



References:
 [1] Burton, Emil, "On Some Convergent Series", <Smarandache Notions Journal>,
     Vol. 7, No. 1-2-3, August 1996, pp. 7-9.



              *
12)  Let f : N  ----> R be a function which satisfies the condition

                                c
           f(t) <= -------------------------------
                      alpha
                     t      (d(t!)) - d((t-1)!)

           for t a nonzero natural number, d(x) the number of divisors of x,
           and the given constants alpha > 1, c > 1.

     Then the series

                     Sigma f(S(n))
                     n>=1

     is convergent to a number s11 . 
                                  f


Reference:
 [1] Burton, Emil, "On Some Convergent Series", <Smarandache Notions 
     Journal>, Vol. 7, No. 1-2-3, August 1996, pp. 7-9.



13)  The series

                              1
                Sigma ------------------
                n>=1     n             n
                      ( Product S(k)! )
                        k=2


     is convergent to a number s13.



Reference:
 [1] Burton, Emil, "On Some Convergent Series", <Smarandache Notions Journal>, 
     Vol. 7, No. 1-2-3, August 1996, pp. 7-9.




14)  The series

                              1
                Sigma -----------------------------, where p > 1,
                n>=1            _____           p
                        S(n)! \/S(n)! (log S(n))
                        


     is convergent to a number s14.



Reference:
 [1] Burton, Emil, "On Some Convergent Series", <Smarandache Notions Journal>, 
     Vol. 7, No. 1-2-3, August 1996, pp. 7-9.




15)  The series

                            n
                           2
                Sigma -------------, 
                n>=1        n
                         S(2  )! 


     is convergent to a number s15.


Reference:
 [1] Burton, Emil, "On Some Convergent Series", <Smarandache Notions Journal>, 
     Vol. 7, No. 1-2-3, August 1996, pp. 7-9.



16)  The series

                       S(n)
                Sigma --------, where p is a real number > 1,
                n>=1     1+p
                        n


    converges to a number s16.
    (For 0 <= p <= 2 the series diverges.)



Reference:
 [1] Burton, Emil, "On Some Convergent Series", <Smarandache Notions Journal>, 
     Vol. 7, No. 1-2-3, August 1996, pp. 7-9.