An advanced necessary test for convergent number series and some consequences
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Інститут проблем штучного інтелекту МОН України та НАН України
2013
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| Цитувати: | An advanced necessary test for convergent number series and some consequences / L.P. Mironenko, I.V. Petrenko // Искусственный интеллект. — 2013. — № 2. — С. 127–131. — Бібліогр.: 3 назв. — англ. |
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| author | Mironenko, L.P. Petrenko, I.V. |
| author_facet | Mironenko, L.P. Petrenko, I.V. |
| citation_txt | An advanced necessary test for convergent number series and some consequences / L.P. Mironenko, I.V. Petrenko // Искусственный интеллект. — 2013. — № 2. — С. 127–131. — Бібліогр.: 3 назв. — англ. |
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ISSN 1561-5359 «Штучний інтелект» 2’2013 127
5М
УДК 514.116
L.P. Mironenko, I.V. Petrenko
Donetsk National Technical University, Ukraine
Ukraine, 83000, Donetsk, Аrtema str., 58, mironenko.leon@yandex.ru
An Advanced Necessary Test For Convergent
Number Series And Some Consequences
Л.П. Мироненко, И.В.Петренко
Донецкий национальный технический университет, Украина
Украина, 83000, г. Донецк, ул. Артема, 58, mironenko.leon@yandex.ru
Усиленный необходимый признак
сходимости рядов и следствия
Л.П. Мироненко, I.В. Петренко
Донецький національний технічний університет, Україна
Україна, 83000, м. Донецьк, вул. Артема, 58
Підсилена необхідна ознака
збіжності рядів та наслідки
In the paper an advanced necessary test for convergence of number series with non-negative terms is
formulated and proved. The test is written in the form 0lim =
∞→
n
n
nu . This formula can be got by using
Dzeta–function and has more possibilities than the usual test 0lim =
∞→
n
n
u . The new test gives new
representations of the necessary test for convergence of non-negative number series.
Keywords: series, convergence, divergence, comparison test, advanced necessary test, limit,
Cauchy’s integral test.
В работе сформулирован и доказан усиленный необходимый признак сходимости числовых рядов с
неотрицательными членами в виде 0lim =
∞→
n
n
nu . Признак получен на основе признака сравнения с
обобщенно гармоническим рядом. Признак имеет более широкие возможности по сравнению с обычным
необходимым признаком сходимости 0lim =
∞→
n
n
u . Применение усиленного необходимого признака
сходимости к интегральному признаку Коши дает новые формы представлений необходимого признака
сходимости рядов с неотрицательными членами.
Ключевые слова: ряд, сходимость, расходимость, признаки сравнения, усиленный необходимый
признак, предел, интегральный признак Коши.
У роботі сформульована і доведена посилена необхідна ознака збіжності чисельних рядів з невiд’ємними
членами у видi 0lim =
∞→
n
n
nu . Ця формула отримана на підставі ознаки порівняння з гармонійним рядом і
має значно більше можливостей у порівнянні з традицiйною необхiдною ознакою збiжностi 0lim =
∞→
n
n
u .
Посилена ознака збіжності разом з інтегральною ознакою Коші відкривають можливість отримати цілу
низку нових представлень ознаки збіжності рядів з невiд’ємними членами.
Ключові слова: ряд, збіжність, ознака порівняння, посилена необхідна ознака, границя,
інтегральна ознака Коші.
Mironenko L.P., Petrenko I.V.
«Искусственный интеллект» 2’2013 128
5М
А
Introduction
Let us recall the content of the necessary test for convergent number series with non-
negative terms. If the series )0( ,
1
≥∑
∞
=
n
n
n
uu converges, then the limit of its common term
tends to zero at ∞→n , i.e. 0lim =
∞→
n
n
u . Usually this test is used to determine the divergence
of series, because it is not sufficient criterion, but is the necessary one [1].
Among sufficient tests, perhaps, the most common is the comparison test, which is
usually used in two forms: in the limit form and in the finite form. In both cases, members
of the investigated series ∑
∞
=1n
n
u are compared with members of the known series∑
∞
=1n
n
v .
We are interested in the limit sufficient test, where
n
n
n
u
ν∞→
lim is considered. If the series
)0( ,
1
>∑
∞
=
n
n
n
vv converges and the value of
n
n
n
u
ν∞→
lim is equal to ∞<C (in a particular
case 0=C ) then the series ∑
∞
=1n
n
u converges. If the series )0( ,
1
>∑
∞
=
n
n
n
vv diverges and
the value of
n
n
n
u
ν∞→
lim is equal to 0>C then the series ∑
∞
=1n
n
u diverges also [1-2].
There are three standard series, which are used in the comparison test. They are the
harmonic one with the common term nv
n
/1= , the generalized harmonic one (so called
zeta function) with the common term
α
n
v
n
1
= and the geometric progression one with the
common term .
n
n
qv =
The limit comparison test with respect to the generalized harmonic series is
considered in the paper. This allows intensify the necessary test for convergent number
series with non-negative terms and expand significantly the possibilities of the usual
necessary test .0lim =
∞→
n
n
u
1 Deduction of the advanced necessary test
for convergent number series
Let us represent the generalized harmonic series in the form
−≤
−>
=
+
∞
=
∑
,0
01
1
1 seriesdivergentfor
seriesconvergentfor
n
n
β
β
β
where instead of the usual parameter α the new parameter β is used: βα +=1 , and
apply the limit comparison test for an arbitrary number series: )0( ,
1
≥∑
∞
=
n
n
n
uu :
n
n
n
n
un
n
u
⋅=
+
∞→
+
∞→
β
β
1
1
lim
/1
lim . If there exists the finite value limit
∞<=⋅
+
∞→
Cun
n
n
1
lim
β (1)
An Advanced Necessary Test For Convergent Number …
«Штучний інтелект» 2’2013 129
5М
and 0>β , then the series )0( ,
1
≥∑
∞
=
n
n
n
uu converges. In other cases at 0≤β the
limit is equal to infinity ( or does not exist) the series )0( ,
1
≥∑
∞
=
n
n
n
uu diverges [1-3].
The test (1) may be represented in the other form if it is applied by
L’Hospital’s rule. Indeed according to the necessary test for convergent number series
0→
n
u at ∞→n , Then ∞→=
−
n
n
u
u
11 at ∞→n and
( ) ( )′
=
′
′
=
∞
∞
==⋅
−
∞→
−
∞→∞→
−
∞→∞→∞→∞→ 11
1
limlimlimlimlimlimlim
n
n
n
nn
n
nn
n
nn
u
n
u
n
n
u
n
nunn
β
βββ .
As the result the test (1) will get the form
( )
∞<=
′
−
∞→
C
u
n
n
n 1
lim
β
(2)
If the series )0( ,
1
>∑
∞
=
n
n
n
uu converges then the equality (1) takes place at 0>β .
The test (1) can be written in the form
∞<=⋅=⋅=
∞→∞→∞→
+
∞→
Cnunnunun
n
nn
n
n
n
n
limlimlimlim
1 βββ . (3)
Suppose that 0lim ≠
∞→
n
n
nu . In this case the equality (3) will be correct, when β
n
n ∞→
lim is
equal to a number at 0>β . But there is no such ,0>β because ∞=
∞→
β
n
n
lim for any 0>β .
Thus, when 0lim ≠
∞→
n
n
nu , then the series ∑
∞
=1n
n
u diverges. So we have the new advanced
necessary test for convergent number series with non-negative terms :)0( ,
1
≥∑
∞
=
n
n
n
uu
0lim =
∞→
n
n
nu . (4)
By the same way the formula (2) is followed the another form of the new advanced
necessary test for convergent number series with non-negative terms :)0( ,
1
≥∑
∞
=
n
n
n
uu
( )
0
1
lim
1
=
′
−
∞→
n
n
u
. (5)
These two forms of the new advanced necessary test for convergent number series
have more opportunities with respect to the casual necessary test for convergent number
series .0lim =
∞→
n
n
u
For example, for the number series∑
∞
=1
1
n
n
the casual necessary test
)0lim( =
∞→
n
n
u
gives 0
1
limlim ==
∞→∞→
n
u
n
n
n
. This means the series must converge, but it
diverges. At the same time according to the formulae (4) and (5) the new advanced
necessary test gives: ∞==
∞→∞→
n
n
nu
n
n
n
limlim
and
( )
.2lim
1
lim
1
∞==
′ ∞→
−
∞→
n
u
n
n
n
It means the
series∑
∞
=1
1
n
n
diverges.
Mironenko L.P., Petrenko I.V.
«Искусственный интеллект» 2’2013 130
5М
А
2 The new advanced necessary test for convergent number series is
applied to Cauchy’s integral test
According to Cauchy’s integral test if the function )(xu is nonnegative and
decreasing at 1≥x then the series ∑
∞
=1
)(
n
nu converges if and only if the improper
integral ∫
+∞
1
)( dxxu
converges.
Let the series ∑
∞
=1
)(
n
nu converges therefore the improper integral ∫
+∞
1
)( dxxu
converges
also. The last is integrated by parts:
.)()1()(lim)(
11
∫∫
+∞
+∞→
+∞
′−−= dxxuxuxxudxxu
x
According to the new advanced necessary test for convergent number series (4)
0)(lim =
+∞→
xxu
x
. Since the left-hand side integral converges then the right-hand side integral
∫
+∞
′
1
)( dxxux must converge also and according to Cauchy’s integral test the series ∑
∞
=
′
1n
n
un
converges.
Take into account, that in this case the advanced necessary test (4) for convergent
number series∑
∞
=
′
1n
n
un takes the form: 0lim
2
=′
∞→
n
n
un . We see that convergence of the
series∑
∞
=
′
1n
n
un is necessary and sufficient condition for convergence of the series∑
∞
=1n
n
u .
Indeed, let us assume that series ∑
∞
=
′
1n
n
un converges therefore the improper integral
∫
+∞
′
1
)( dxxux converges also. The last is integrated by parts:
.)()1()(lim)(
11
∫∫
+∞
+∞→
+∞
−−=′ dxxuuxxudxxux
x
Since 0)(lim =
+∞→
xxu
x
is necessary and sufficient condition for convergence of the
series∑
∞
=1n
n
u , then the series ∑
∞
=1n
n
u converges. If the condition 0)(lim =
+∞→
xxu
x
is not
satisfied then the series ∑
∞
=1n
n
u diverges and the series ∑
∞
=
′
1n
n
un diverges also.
Let us apply this reasoning to the series ∑
∞
=
′
1
)(
n
nun , given that 0lim
2
=′
∞→
n
n
un is
necessary and sufficient condition for convergence of the series∑
∞
=
′
1
)(
n
nun . Let the series
∑
∞
=
′
1
)(
n
nun converges therefore the improper integral
∫
+∞
′
1
)( dxxux
converges also. The last is
integrated by parts:
An Advanced Necessary Test For Convergent Number …
«Штучний інтелект» 2’2013 131
5М
( ) .)(
2
1
)1()(lim
2
1
)(
1
22
1
∫∫
+∞
+∞→
+∞
′′−′−′=′ dxxuxuxuxdxxux
x
Let 0lim
2
=′
∞→
n
n
un . Since the left-hand side integral converges then the right-hand side
integral ∫
+∞
′′
1
2
)( dxxux converges also and according to Cauchy’s integral test the series
∑
∞
=
′′
1
2
)(
n
nun converges. These arguments can be repeated k times. As result two important
statements are obtained.
1. In order to the series with non-negative terms ∑
∞
=1
)(
n
nu
converges it is necessary
and sufficient, that the series ∑
∞
=1
)(
)(
n
kk
nun , ,...)2,1( =k
converges too.
2. The series with non-negative terms ∑
∞
=1
)(
n
nu
converges if and only if the advanced
necessary test in the form 0)(lim )(1
=
+
+∞→
nun
kk
n
, ,...)2,1( =k
takes place.
These statements can be demonstrated by the example of the convergent
series∑
∞
=1
2
ln
1
n
xn
with the common term .
ln
1
2
xn
u
n
= If the common term is differentiated,
then
.
ln
2
ln
6ln6ln2
,
ln
1
ln
2ln
2343
2
2332
nnnn
nn
u
nnnn
n
u
n
n
n
n
∞→∞→
→
++
=′′−→
−−
=′
The series ∑
∞
=1
)(
)(
n
kk
nun ( 2,1=k ) have the next forms:
nn
n
nun
nn
3
11 ln
2ln
)(
−−
=′ ∑∑
∞
=
∞
=
, .
ln
6ln6ln2
)(
1
4
2
1
2
∑∑
∞
=
∞
=
++
=′′
nn
nn
nn
nun
For sufficiently large values of n the common term of each of the series behaves like
the common term of the initial series, namely .
ln
1
2
2
nn
ununu
n
nnn
∞→
→′′≈′≈
Findings
1. An advanced necessary test for convergence of number series with non-negative
terms is formulated and proved. The test is much more powerful than the casual necessary test.
2. The advanced necessary test gives rich possibilities to get equivalent converged
series with respect to the given series. These series cannot be obtained by applying the
usual way in the series theory.
Literature
1. Kudriavtsew L.D. Маtematichesky analiz. Том I., Nаukа,1970.– 571 p.
2. Ilyin V.А., Pozdniak E.G. Оsсnowy mаtematicheskogo analiza, tом 1, Izd. FML, Моskwa, 1956. – 472 p.
3. Fikhtengolts G.М. Кurs differentsialnogo I integralnogo ischislenia, tом 1, Nauka, «FML», 1972 – 795 p.
The article received 02.04.2013.
|
| id | nasplib_isofts_kiev_ua-123456789-85214 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1561-5359 |
| language | English |
| last_indexed | 2025-11-25T20:53:18Z |
| publishDate | 2013 |
| publisher | Інститут проблем штучного інтелекту МОН України та НАН України |
| record_format | dspace |
| spelling | Mironenko, L.P. Petrenko, I.V. 2015-07-21T19:17:26Z 2015-07-21T19:17:26Z 2013 An advanced necessary test for convergent number series and some consequences / L.P. Mironenko, I.V. Petrenko // Искусственный интеллект. — 2013. — № 2. — С. 127–131. — Бібліогр.: 3 назв. — англ. 1561-5359 https://nasplib.isofts.kiev.ua/handle/123456789/85214 514.116 en Інститут проблем штучного інтелекту МОН України та НАН України Искусственный интеллект Обучающие и экспертные системы An advanced necessary test for convergent number series and some consequences Підсилена необхідна ознака збіжності рядів та наслідки Усиленный необходимый признак сходимости рядов и следствия Article published earlier |
| spellingShingle | An advanced necessary test for convergent number series and some consequences Mironenko, L.P. Petrenko, I.V. Обучающие и экспертные системы |
| title | An advanced necessary test for convergent number series and some consequences |
| title_alt | Підсилена необхідна ознака збіжності рядів та наслідки Усиленный необходимый признак сходимости рядов и следствия |
| title_full | An advanced necessary test for convergent number series and some consequences |
| title_fullStr | An advanced necessary test for convergent number series and some consequences |
| title_full_unstemmed | An advanced necessary test for convergent number series and some consequences |
| title_short | An advanced necessary test for convergent number series and some consequences |
| title_sort | advanced necessary test for convergent number series and some consequences |
| topic | Обучающие и экспертные системы |
| topic_facet | Обучающие и экспертные системы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/85214 |
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