ON SOME SERIES INVOLVING SMARANDACHE FUNCTION by
Emil Burton
The study of infinite series involving Smarandache function is one of the most interesting aspects of analysis. In this brief article me give only a bare introduction to it.
E
First we prove that the series
K=2
S(k) (kH) !
converges and has
OE]e- 22' ~r2 l
the sum
S (m) is the Smarandache function: Let
us n
~
L.J ic=2
1:' 1! 2!
denote S(k) (k+1)!
<
1 n!
1 + - + - + ... + -
1
2
S(m)
= min
by
=
k=2
n
k
implies that
show
that
as follows:
n
~
We
En'
k E--,------;--,--(k+1)!
S(k)
{kEN;.mlk!}
E
k=2
S(k) (k+1) !
13
~
E . .(k+l)! .,. . k,. .---,-n
k=2
=
1
1 (n+l) !
2!
1
1
2
(k+l) !
<
1 2
On the other consequently:
hand
S (k)
r.
~
>
(k+l)!
~
ic=Z
S(k)
=
(k+:)!
5 E::+1 -2 <
It follows that
L
1
n
implies
k~2
-
~
3!
4!
~+-==-+
n
~
~
1 2
<
S(k) +1 ,
S(k)
~
2
S{k}
We can also check that
=
and a
> -
S (k)
and
5 2
E~.,-Â
.â&#x20AC;˘
~
therefore
Ef e- 22 L
I
1 -
2
are strictly and
k
OE]e-~1 ~[ .
Hence
.
~
S(k)
I
n+l'
is a convergent series with sum
(k+l) !
REMARK: Some of inequalities k
+
...
S(k) (k+l) !
E k-2
that
rENe
(k-I) !
S(k} (k+r) !
and
rEN,
are both convergent as follows:
t
k=r
S(k) (k-r)!
~
Zl
L
.:.. = r(-2-+~+ ... + (n-r) O! 1! n
We
get
L ie-r
r +2
_r-:+--,-(n_--:-I __ }
- + - - + - - + ... + O! l! 2! (n- I) !
(k-r)!
k=r
I +1
I
k
. + (1 2
-~-+
I!
! )
2!
...
+
S{k)
n-r ) (n-I ) ! = IEn_r
=
+ En - r - 1
which
(k-r) !
converges. S(k)
Also we have
(k+r) !
Let us define the set If
So,
m=
-
L
m=3
00
n! 2
-.
mEM.z
m
=:xl
S{m)!
k=2
IEN .
~ = {mEN: m
-
m SCm}
!
=
~! neN, n ~ 3} . I
test
=
n! 2 n!
and therefore
lDEHz
A problem:
L
I
it is obvious that
mEM.z
SCm) = n,
<
the
L
k=2 kEN
k
S(k}!
=
00
convergence behaviour of
1 S(k) !
kEN
,.
the
series
R~FERENCES
1. Smarandache Function Journal, Vol.1 1990, Vol. 2-3, 1993, Vol.4-5 1994, Number Theory Publishing, Co., R. Muller Editor, Phoenix, New York, Lyon.
Curent Address: Dept. of Math. University of Craiova, Craiova (1100)
- Romania