SMARANDACHE SERIES
by Jose Castillo, Navajo Community College,
Tsaile, Arizona, USA
A Smarandache Series is any series which involves at least a Smarandache
type Sequence, Sub-Sequence, or Function.
(Over 200 such sequences, subsequences, and functions are listed in Sloane's
database of Encyclopedia of Integer sequences -- online).
For examples:
a) If we consider the smarandache consecutive sequence:
1, 12, 123, 1234, 12345, ..., 123456789101112, ...
we form a smarandache series:
1/1 + 1/12 + 1/123 + 1/4321 + ... .
b) If we attach the smarandache reverse sequence:
1, 21, 321, 4321, 54321, ..., 121110987654321, ...
to the previous one we get another smarandache series:
1/1 + 12/21 + 123/321 + 1234/4321 + ... .
With a mathematics software it is possible to calculate such series
to see which ones of them converge, and eventually to make
conjectures, or to algebraically prove those converging towards certain
constants.
The first series is convergent, while the second one is divergent
(Ashbacher[1]).
Reference:
[1] Ashbacher, Charles, "Smarandache Series Convergence", to appear.
[2] Smarandoiu, Stefan, "Convergence of Smarandache Continued Fractions",
,
Vol. 17, No. 4, Issue 106, 1996, p. 680.
[3] Zhong, Chung, "On Smarandache Continued Fractions", , Vol. 9, No. 1-2, 1998, pp. 40-42.