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