Classification of the necessary tests of convergence in the theory of numerical series
The paper suggests three types of the necessary tests for the convergence of numerical series with positive terms. It is shown that these tests are consequences of the limit test of comparison with respect to the standard series: harmonic and logarithmic ones. The new tests have different capabi...
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irk-123456789-852202015-07-23T03:02:06Z Classification of the necessary tests of convergence in the theory of numerical series Mironenko, L.P. Petrenko, I.V. Vlasenko, A.Yu. Интеллектуальные системы планирования, управления, моделирования и принятия решений The paper suggests three types of the necessary tests for the convergence of numerical series with positive terms. It is shown that these tests are consequences of the limit test of comparison with respect to the standard series: harmonic and logarithmic ones. The new tests have different capabilities and can be applied to almost any series, whose terms form a monotone sequence. У роботі запропоновані три види необхідних ознак збіжності числових рядів з додатними членами. Показано, що ці ознаки є наслідками застосування граничного ознаки порівняння зі стандартними рядами: гармонічним та логарифмічним. Нові ознаки збіжності мають різні можливості і можуть застосовуватися практично до будь-яких рядах, члени яких утворюють монотонну послідовність. В работе предложены три вида необходимых признаков сходимости числовых рядов с положительными членами. Показано, что эти признаки являются следствиями применения предельного признака сравнения к стандартным рядам: гармоническому и логарифмическому. Новые признаки сходимости имеют различные возможности и могут применяться практически к любым рядам, члены которых образуют монотонную последовательность. 2013 Article Класифікація необхідних умов збіжності в теорії числових рядів / L.P. Mironenko, I.V. Petrenko, A.Yu Vlasenko // Искусственный интеллект. — 2013. — № 3. — С. 294–299. — Бібліогр.: 6 назв. — англ. 1561-5359 http://dspace.nbuv.gov.ua/handle/123456789/85220 514.116 en Искусственный интеллект Інститут проблем штучного інтелекту МОН України та НАН України |
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Интеллектуальные системы планирования, управления, моделирования и принятия решений Интеллектуальные системы планирования, управления, моделирования и принятия решений |
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Интеллектуальные системы планирования, управления, моделирования и принятия решений Интеллектуальные системы планирования, управления, моделирования и принятия решений Mironenko, L.P. Petrenko, I.V. Vlasenko, A.Yu. Classification of the necessary tests of convergence in the theory of numerical series Искусственный интеллект |
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The paper suggests three types of the necessary tests for the convergence of numerical series with positive
terms. It is shown that these tests are consequences of the limit test of comparison with respect to the
standard series: harmonic and logarithmic ones. The new tests have different capabilities and can be applied
to almost any series, whose terms form a monotone sequence. |
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Mironenko, L.P. Petrenko, I.V. Vlasenko, A.Yu. |
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Mironenko, L.P. Petrenko, I.V. Vlasenko, A.Yu. |
| author_sort |
Mironenko, L.P. |
| title |
Classification of the necessary tests of convergence in the theory of numerical series |
| title_short |
Classification of the necessary tests of convergence in the theory of numerical series |
| title_full |
Classification of the necessary tests of convergence in the theory of numerical series |
| title_fullStr |
Classification of the necessary tests of convergence in the theory of numerical series |
| title_full_unstemmed |
Classification of the necessary tests of convergence in the theory of numerical series |
| title_sort |
classification of the necessary tests of convergence in the theory of numerical series |
| publisher |
Інститут проблем штучного інтелекту МОН України та НАН України |
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2013 |
| topic_facet |
Интеллектуальные системы планирования, управления, моделирования и принятия решений |
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http://dspace.nbuv.gov.ua/handle/123456789/85220 |
| citation_txt |
Класифікація необхідних умов збіжності в теорії числових рядів / L.P. Mironenko, I.V. Petrenko, A.Yu Vlasenko // Искусственный интеллект. — 2013. — № 3. — С. 294–299. — Бібліогр.: 6 назв. — англ. |
| series |
Искусственный интеллект |
| work_keys_str_mv |
AT mironenkolp classificationofthenecessarytestsofconvergenceinthetheoryofnumericalseries AT petrenkoiv classificationofthenecessarytestsofconvergenceinthetheoryofnumericalseries AT vlasenkoayu classificationofthenecessarytestsofconvergenceinthetheoryofnumericalseries |
| first_indexed |
2025-07-06T12:23:10Z |
| last_indexed |
2025-07-06T12:23:10Z |
| _version_ |
1836900250795638784 |
| fulltext |
ISSN 1561-5359 «Искусственный интеллект» 2013 № 3 294
4M
UDK 514.116
L.P. Mironenko, I.V. Petrenko, A.Yu. Vlasenko
Donetsk National Technical University,
Ukraine, 83000, Donetsk, Аrtema str., 58
Classification of the Necessary Tests
of Convergence in the Theory of Numerical Series
Л.П. Мироненко, И.В. Петренко, А.Ю. Власенко
Донецкий национальный технический университет, Украина
Украина, 83000, г.Донецк, ул. Артема, 58
Классификация необходимых условий сходимости
в теории числовых рядов
Л.П. Мироненко, I.B. Петренко, А.Ю. Власенко
Донецький національний технічний університет, Україна
Україна, 83000, м. Донецьк, вул. Артема, 58
Класифікація необхідних умов збіжності
в теорії числових рядів
The paper suggests three types of the necessary tests for the convergence of numerical series with positive
terms. It is shown that these tests are consequences of the limit test of comparison with respect to the
standard series: harmonic and logarithmic ones. The new tests have different capabilities and can be applied
to almost any series, whose terms form a monotone sequence. The obtained necessary tests for the
convergence of numerical series have more opportunities, than the accepted standard one .0lim =
∞→
n
n
u
Keywords: limit, L’Hospital’s rule, series, convergence, divergence, test of comparison of series,
harmonic series, logarithmic series and geometric progression series.
В работе предложены три вида необходимых признаков сходимости числовых рядов с положительными
членами. Показано, что эти признаки являются следствиями применения предельного признака сравнения
к стандартным рядам: гармоническому и логарифмическому. Новые признаки сходимости имеют различные
возможности и могут применяться практически к любым рядам, члены которых образуют монотонную
последовательность. Полученные признаки имеют более широкие возможности, чем стандартный
необходимый признак сходимости рядов .0lim =
∞→
n
n
u
Ключевые слова: предел, правило Лопиталя, сходимость, расходимость, признак сравнения
рядов, гармонический ряд, логарифмический ряд, ряд геометрической прогрессии.
У роботі запропоновані три види необхідних ознак збіжності числових рядів з додатними членами.
Показано, що ці ознаки є наслідками застосування граничного ознаки порівняння зі стандартними рядами:
гармонічним та логарифмічним. Нові ознаки збіжності мають різні можливості і можуть застосовуватися
практично до будь-яких рядах, члени яких утворюють монотонну послідовність. Отримані ознаки мають
більш широкі можливості, ніж стандартна необхідна ознака збіжності рядів .0lim =
∞→
n
n
u
Ключові слова: границя, правило Лопіталя, збіжність, розбіжність, ознака порівняння рядів,
гармонічний ряд, логарифмічний ряд, ряд геометричної прогресії.
Classification of the Necessary Tests of Convergence in the Theory of Numerical Series
«Штучний інтелект» 2013 № 3 295
4M
Introduction
The necessary tests for the convergence of series play an important practical role in
the evaluation of the convergence and divergence of numerical series with non-negative
terms. It is believed that they are negative, because they can only define the divergence of
the series. If the necessary tests are violated, we can definitely affirm the divergence of the
series. If they are satisfied, the question of convergence can be finally resolved only by
sufficient tests for the convergence of series [1]. First of all, these are tests of comparison
with the standard series, such as the series of geometric progression, generalized harmonic
series and generalized logarithmic series [2].
In the theory of the numerical series the test of comparison of series with positive
terms commonly used in two forms - in the finite form and in the limit form [3]. In the first
case, the two series are compared. The first series is
0 ,
1
≥∑
∞
=
n
n
n
uu
(1)
and the second series is 0 ,
1
≥∑
∞
=
n
n
n
vv . If there is a number 0>M such that, starting from a
certain numberN , the inequality
nn
vMu ⋅≤ takes place and the series ∑
∞
=1n
n
v converges, then
the series ∑
∞
=1n
n
u converges too. But if the series ∑
∞
=1n
n
u diverges, then the series ∑
∞
=1n
n
v
diverges also. [1]
In the limit test of comparison the expression
nn
n
vu /lim
∞→
is considered. If the series
0 .
1
>∑
∞
=
n
n
n
vv converges, and the limit value is equal to ∞<C or 0C = , then the series
∑
∞
=1n
n
u converges also. If the series ∑
∞
=1n
n
v diverges, and the size of the limit is equal to
0C ≠ orC =∞ , then the series ∑
∞
=1n
n
u diverges also [3], [4].
The series of comparison is usually taken from the three standard series: the harmonic
one with the general term nv
n
/1= , the generalized harmonic one with the general term
α
nv
n
/1= and the series of geometric progression with the general term
n
n
qv = .
1 The Necessary Test for the Convergence of Series
in the Classical Approach
Theorem. If the series (1) 0 ,
1
≥∑
∞
=
nn n
uu converges, then the limit of the general term of
the series is zero
.0lim =
∞→
n
n
u
(2)
Proof. If the series (1) converges, then there is a finite limit of the sequence of partial
sums ∑
=
=
k
n nk
us
1
, where ss
k
k
=
∞→
lim . The same limit is the limit of the sequence of partial
sums
1
{ }
k
s
−
, where ss
k
k
=
−
∞→
1
lim . Then we have ( ) 0limlim 1 =−=−=
−
∞→∞→
ssssu
nn
n
n
n
.
Consequence. The series 0 ,
1
≥∑
∞
=
nn n
uu diverges at 0lim ≠
∞→
n
n
u .
Remark. The condition 0lim =
∞→
n
n
u is not the sufficient condition for the convergence
of series (1), so it is usually used to establish the divergence of the series.
Mironenko L.P., Petrenko I.V., Vlasenko A.Yu.
«Искусственный интеллект» 2013 № 3 296
4M
Examples.
1. The series
1
1
2 1
n
n
n
∞
=
−
+
∑ diverges, since 0
2
1
1
1
lim ≠=
+
−
∞→ n
n
n
.
2. The series
2
1
1
(2 1)
n
n
n
∞
=
−
+
∑ is given. The necessary test (2) of convergence is perfor-
med, since 0
)12(
1
lim
2
=
+
−
∞→ n
n
n
, but the conclusion about the convergence of the series cannot
be done. It is necessary to use any sufficient test of convergence.
3. The series
2
1
1
(2 1) ln( 1)
n
n
n n
∞
=
−
+ +
∑ is given. The necessary test (2) of convergence is
performed, since 0
)1ln()12(
1
lim
2
=
++
−
∞→ nn
n
n
. As in the previous example, the conclusion about
the convergence of the series cannot be done.
2 The Necessary Test for the Convergence of Series
Compared with Divergent Series
1. The necessary test of convergence of the series in comparison with the harmonic
series. Let us write the limit test of comparison of any arbitrary series (1) with respect to
the harmonic series
1
, , 1/ :
n n
n
v v n
∞
=
=∑ n
n
n
n
n
n
n
un
n
u
v
u
⋅==
∞→∞→∞→
lim
/1
limlim .
Consider all the possible values of this limit.
lim lim 0
0
n
n
n n
n
u
n u С
v→∞ →∞
∞
= ⋅ = ≠
(3)
It is known that the harmonic series diverges. If this limit is equal to ∞or 0С ≠ , then
the series (1) diverges. If the limit is equal to 0, then the series (1) either converges or
diverges. So the test of comparison of series in the limit form works. Let us change the
statement of the problem. Now assume that the series (1) converges. Then the first two
values of the limit (3) ∞or 0С ≠ are not possible, because they disagree to the assumption
that the series (1) converges. So remains only the third option, which is equal to zero. This
case should be understood as the necessary test for convergence of the series (1).
0lim =⋅
∞→
n
n
un . (4)
Thus, it has been established from the limit comparison test with respect to the
harmonic series, that the new necessary test for the convergence of the series in the form
(4) is much "stronger" than the standard one in the form (2).
Examples.
2. The series
2
1
1
(2 1)
n
n
n
∞
=
−
+
∑ is given. The necessary test for the convergence of the series in
the form (4) is not performed, since
2
1 1
lim .
(2 1) 4n
n
n
n→∞
−
⋅ =
+
Hence the series diverges. Above
in Example 2, when the necessary test (2) was applied to the same series, the conclusion of
convergence of the series was impossible.
Classification of the Necessary Tests of Convergence in the Theory of Numerical Series
«Штучний інтелект» 2013 № 3 297
4M
3. The series∑
∞
=
++
−
1
2 )1ln()12(
1
n
nn
n
is given. The necessary test for the convergence of
the series in the form (4) is performed, since 0
)1ln()12(
1
lim
2
=
++
−
∞→ nn
n
n
n
, but the conclusion
of the convergence of this series cannot be done.
Remark. The necessary test (4) works only for series, whose terms form a monotone
sequence, i.e.
1+
>
nn
uu . Otherwise the limit (4) may not exist. For example, in the series
1 n
n
u
∞
=
∑ with the general term
≠
=
=
2
22
0
/1
mnat
mnatm
u
n
the terms
n
u do not form a monotone
sequence. For this series the limit (4) does not exist.
4. The necessary test of convergence of the series in comparison with the
logarithmic series. Let us write the limit test of comparison of any arbitrary series (1) with
respect to the logarithmic series
1
, , 1/ ln :
n n
n
v v n n
∞
=
=∑
n
n
n
n
n
n
n
unn
nn
u
v
u
⋅==
∞→∞→∞→
lnlim
ln/1
limlim . (5)
Repeat the same reasoning as for the harmonic series. It is known that this logarithmic
series diverges. Suppose that the series (1) converges. The values ∞ or 0С ≠ of the limit (5)
are not possible, since they contradict to the assumption of convergence of the series (1).
Remains only one option - the limit (5) is zero. The latter case is to be understood as the
necessary test for the convergence of the series (1):
0lnlim =⋅⋅
∞→
n
n
unn . (6)
Thus, from the limit comparison test of series was established the second new necessary
test for the convergence of series in the form (6), which is “stronger”, than test (4).
Example.
5. The series ∑
∞
=
++
−
1
2 )1ln()12(
1
n
nn
n
is given. The necessary test for the convergence of
the series in the form (6) is not performed, since
4
1
)1ln()12(
1
lnlim
2
=
++
−
∞→ nn
n
nn
n
. Hence the
series diverges. Above, in Examples 2 and 5, the necessary tests (2) and (4) were applied,
but the conclusion of the convergence of series could not be done.
3 The Other Representations of Necessary Tests
for Convergence Of Series
Let us designate the series (1) thus
nn
n
n
uw
w
/1,
1
1
=∑
∞
=
. Then the first necessary test
for the convergence of the series (2) will be:
0
1
lim =
∞→
n
n w
; ∞=
∞→
n
n
wlim . (7)
So the second necessary test for convergence of series (4) will be 0lim =
∞→
n
n w
n
. From
Mironenko L.P., Petrenko I.V., Vlasenko A.Yu.
«Искусственный интеллект» 2013 № 3 298
4M
the necessary test (2) it follows ∞→
n
w . Therefore, there is indeterminacy }/{ ∞∞ , which
will be disclosed by L’Hospital’s rule
n
n
n
n
n
n ww
n
w
n
′
=
′
′
=
∞→∞→∞→
1
limlimlim . As a result, we came to
the modification of the second necessary test for convergence of series in the form:
0
1
lim =
′∞→
n
n w
или .lim ∞=′
∞→
n
n
w
(8)
The third necessary test for convergence of series (4) will be 0
ln
lim =
∞→
n
n w
nn
. From the
necessary test (2) it follows ∞→
n
w .Then there is indeterminacy }/{ ∞∞ , which can be
disclosed by L’Hospital’s rule
( )lnln 1 1
lim lim lim lim lim lim 0
n n n n n n
n n n n
nn n
n n
w w nw w→∞ →∞ →∞ →∞ →∞ →∞
′
= = = =
′ ′ ′
. As a
result it is obtained that the third modified necessary test for convergence of series coin-
cides with the second modified necessary test for convergence of series (8). However, this
third necessary test for convergence of series can be expressed via the second derivative:
( )
.
1
lim
ln
lim
ln
lim
1
limlnlimlimlnlim
ln
lim
n
n
n
n
n
n
n
nn
n
nn
n
n wnw
n
w
n
w
n
w
n
n
w
nn
′′
=
′′
′
=
′
=
′
=
′
′
=
∞→∞→∞→∞→∞→∞→∞→∞→
As a result we came to the modification of the third necessary test for convergence of
series in the form: 0
1
lim =
′′∞→
n
n wn
or ∞=′′
∞→
n
n
wnlim .
The obtained results can be tabulated (Table).
Table 1 – Classification of the necessary conditions for convergence of series
with positive terms
A comparison
series
1 n
n
u
∞
=
∑
Necessary test Other forms of necessary test
1
1/
n
n
w
∞
=
∑
Necessary test
opportunities
∑
∞
=1
1
n
0lim =
∞→
n
n
u ∞=
∞→
n
n
wlim light
Harmonic series
∑
∞
=1
/1
n
n
0lim =
∞→
n
n
nu
0
1
lim =
′∞→
n
n w
или ∞=′
∞→
n
n
wlim medium
Logarithmic series
∑
∞
=1
ln/1
n
nn
0lnlim =⋅
∞→
n
n
unn
0
1
lim =
′∞→
n
n w
или 0
1
lim =
′′∞→
n
n wn
strong
In Table 1 the notions light, medium, strong are relative because intermediate
variants and the expansion of the table to the strongest notation are possible. For example,
the logarithmic series of the second order creates stronger necessary test for the conver-
gence of series ( ) 0lnlnlnlim =⋅⋅⋅
∞→
n
n
unnn , than 0lnlim =⋅⋅
∞→
n
n
unn .
Summarizing the results of this paper, it is possible to formulate the theorem on the
necessary condition for the convergence of numerical series with positive terms.
Theorem. If the series
1
, 0,
n n
n
u u
∞
=
≥∑ 1+
>
nn
uu converges, and 0 ,
1
>∑
∞
=
n
n
n
vv is an
Classification of the Necessary Tests of Convergence in the Theory of Numerical Series
«Штучний інтелект» 2013 № 3 299
4M
arbitrary divergent series, then the condition 0lim =
∞→
n
n
n v
u
is the necessary condition for the
convergence of the initial series∑
∞
=1n
n
u .
If the general term of the series
n
v to take equal to1 , n/1 or nn ln/1 it will give us the
necessary conditions for the convergence of the series presented in the table (Table 1).
Findings
This paper describes three types of the necessary tests of convergence of numerical
series with positive terms. Thus
1. It is shown that these tests are the consequences of the application of the limit test
of comparison to the standard series: geometric, harmonic and logarithmic ones.
2. These new necessary tests of the convergence of numerical series have different
capabilities and can be applied to almost any series with positive terms, whose terms form
a monotone sequence.
3. The obtained necessary tests have more opportunities than the accepted standard
necessary test of convergence of numerical series (2).
4. It is shown that the necessary test for the convergence of numerical series in the
standard form 0lim =
∞→
n
n
u can be successfully replaced by more powerful new necessary tests
for the convergence of numerical series, which are the next: 0lim =
∞→
n
n
nu , 0lnlim =⋅
∞→
n
n
unn ,
( ) 0lnlnlnlim =⋅⋅⋅
∞→
n
n
unnn etc.
5. These results can be transferred to the improper integrals.
Literatura
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Literature
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The paper is received by the edition 26.04.2013.
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