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Dinh Van Ti$p vd Dtg Tap chi KHOA HOC & CONG NGHE 162(02): 177-182

THE CONVERGENCE OF A SUB-SERIES OF HARMONIC SERIES

Dinh Van Tiep , Pham Thi Thu Hang

College ofTechnoiogy-TNU

ABSTRACT

When studymg the convergence of a numeric series, one important technique we often use is to

compare that series with a series whose each term is a power of the reciprocal of an integer We

sometimes call this technique p-test. In general, we often estimate that series with one of sub-series

(these are series whose each term is an integer) of the harmonic series. Besides, to test the

convergence of Riemann Zeta function at a given point, by comparing this series with a such sub￾series, since then we may know whether the fimction is defined at this point or not. Therefore,

finding the conditions in which such sub-series converges becomes a meaningful work. This

article aims to present new result for this problem

Keyword: sub-series of the harmonic series, Riemann Zeta function, convergent, numeric series,

the reciprocal of an integer.

INTRODUCTION

We know that the harmonic series / ^ —

n

THE RATE OF THE INCREASING OF

TERMS FOR A DIVERGENT SUB

SERIES

We first consider {"A}^^, to be an increasing

sequence of positive integers. It naturally

establishes the series V — , we index this by

(I). To find out the rate at which each term

increases, we compare this series with

^ 1

convergent series in the form 2 ^ —

( « > 1), and we get the following statement.

Proposition 1. Ifth e series ^ — diverges.

divergent. However, there are hs convergent

sub-series, such as zl'T'ZllJ' ^^ ^^^^ ^

ASl ^ iai ^

general and useful result which we often

consider to use first to test the convergence of

a given series, namely, the series /^- —

converges if and only if p>l and diverges if

P<1. This method is called the p-test. Now,

consider a sub-series / — of the harmonic

series. If this series converges, whether the

sequence {"ij^^, increases at a higher speed

than some sequence \k"\ ,{a>\) does.

This question is answered in Proposhion 1 ,. • , ,.1.

below Besides, if we look at the distance of ^"PP"^^ by contradiction that there exist

two consecutive denominators, «> " Md «> ' '''"=1' *^ '

\-=n,u-n,, we also want to know how liniinf,^> £ >0 . This means, there has an

different it is between this distance with that *"*" ^

distance of the series T— , the main result index k„>l such that — -~jT ' ^°' " "

is stated in Theorem 1. k>k^. This leads to the conclusion that (1)

must be convergent, we meet the

then lim inf - ^ = 0 for all a>\.

This statement is easy to verify. Indeed, we

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