ON SOME SERIES INVOLVING SMARANDACHE FUNCTION

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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~.,-­

.•

~

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


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