The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory of numerical series
In the paper it is shown that D’alembert’s test and Cauchy’s radical test are not independent one from another. It is proposed a transition scheme from one test to another and vice versa. The equivalence of the tests is proved for series with the monotonically decreasing terms. This fact is used t...
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Mironenko, L.P. 2015-07-21T12:21:43Z 2015-07-21T12:21:43Z 2013 The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory of numerical series / L.P. Mironenko // Искусственный интеллект. — 2013. — № 3. — С. 507–511. — Бібліогр.: 5 назв. — англ. 1561-5359 https://nasplib.isofts.kiev.ua/handle/123456789/85175 514.116 In the paper it is shown that D’alembert’s test and Cauchy’s radical test are not independent one from another. It is proposed a transition scheme from one test to another and vice versa. The equivalence of the tests is proved for series with the monotonically decreasing terms. This fact is used to formulate a new test of the convergence for series with positive terms. The test is equivalent to Dalembert and Cauchy’s radical tests, but it has some advantages. It can be applied to any series within Cauchy-D’alembert’s theory. У статті показано, що ознака Даламбера і радикальна ознака Коші не є незалежними одна від другої. Запропоновано схему переходу від однієї ознаки до другої. Еквівалентність ознак доведена для рядів з монотонно регресними членами. Цей факт використан для формулювання нової ознаки збіжності рядів з додатними членами. Нова ознака еквівалентна ознакам Даламбера і Коші, але має деяку перевагу вона може застосовуватися до будь-яких рядів в рамках теорії Коші-Даламбера. В статье показано, что признак Даламбера и радикальный признак Коши не являются независимыми друг от друга. Предложена схема перехода от одного признака к другому. Эквивалентность признаков доказана для рядов с монотонно убывающими членами. Этот факт использован для формулировки нового признака сходимости рядов с положительными членами. Новый признак эквивалентен признакам Даламбера и Коши, но имеет некоторые преимущества – он может применяться к любым рядам в рамках теории Коши-Даламбера. en Інститут проблем штучного інтелекту МОН України та НАН України Искусственный интеллект Обучающие и экспертные системы The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory of numerical series Доказ еквівалентності ознак Даламбера і Коші в теорії числових рядів Доказательство эквивалентности признаков Даламбера и Коши в теории числовых рядов Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory of numerical series |
| spellingShingle |
The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory of numerical series Mironenko, L.P. Обучающие и экспертные системы |
| title_short |
The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory of numerical series |
| title_full |
The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory of numerical series |
| title_fullStr |
The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory of numerical series |
| title_full_unstemmed |
The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory of numerical series |
| title_sort |
proof of an equivalence of d’alembert’s and cauchy’s tests in the theory of numerical series |
| author |
Mironenko, L.P. |
| author_facet |
Mironenko, L.P. |
| topic |
Обучающие и экспертные системы |
| topic_facet |
Обучающие и экспертные системы |
| publishDate |
2013 |
| language |
English |
| container_title |
Искусственный интеллект |
| publisher |
Інститут проблем штучного інтелекту МОН України та НАН України |
| format |
Article |
| title_alt |
Доказ еквівалентності ознак Даламбера і Коші в теорії числових рядів Доказательство эквивалентности признаков Даламбера и Коши в теории числовых рядов |
| description |
In the paper it is shown that D’alembert’s test and Cauchy’s radical test are not independent one from another. It
is proposed a transition scheme from one test to another and vice versa. The equivalence of the tests is proved
for series with the monotonically decreasing terms. This fact is used to formulate a new test of the
convergence for series with positive terms. The test is equivalent to Dalembert and Cauchy’s radical tests, but
it has some advantages. It can be applied to any series within Cauchy-D’alembert’s theory.
У статті показано, що ознака Даламбера і радикальна ознака Коші не є незалежними одна від другої.
Запропоновано схему переходу від однієї ознаки до другої. Еквівалентність ознак доведена для рядів з
монотонно регресними членами. Цей факт використан для формулювання нової ознаки збіжності рядів з
додатними членами. Нова ознака еквівалентна ознакам Даламбера і Коші, але має деяку перевагу вона
може застосовуватися до будь-яких рядів в рамках теорії Коші-Даламбера.
В статье показано, что признак Даламбера и радикальный признак Коши не являются независимыми
друг от друга. Предложена схема перехода от одного признака к другому. Эквивалентность признаков
доказана для рядов с монотонно убывающими членами. Этот факт использован для формулировки
нового признака сходимости рядов с положительными членами. Новый признак эквивалентен признакам
Даламбера и Коши, но имеет некоторые преимущества – он может применяться к любым рядам в
рамках теории Коши-Даламбера.
|
| issn |
1561-5359 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/85175 |
| citation_txt |
The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory of numerical series / L.P. Mironenko // Искусственный интеллект. — 2013. — № 3. — С. 507–511. — Бібліогр.: 5 назв. — англ. |
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| first_indexed |
2025-11-24T23:27:17Z |
| last_indexed |
2025-11-24T23:27:17Z |
| _version_ |
1850500279275880448 |
| fulltext |
ISSN 1561-5359 «Штучний інтелект» 2013 № 3
507
6М
UDK 514.116
L.P.Mironenko
Donetsk National Technical University,
Ukraine, 83000, Donetsk, Аrtema st., 58
The proof of an equivalence of D’alembert’s and
Cauchy’s tests in the theory of numerical series
Л.П. Мироненко
Донецкий национальный технический университет,
Украина, 83000, г. Донецк, ул. Артема, 58
Доказательство эквивалентности признаков Даламбера и
Коши в теории числовых рядов
Л.П. Мироненко
Донецький національний технічний університет,
Україна, 83000, м. Донецьк, вул. Артема, 58
Доказ еквівалентності ознак Даламбера і Коші в теорії
числових рядів
In the paper it is shown that D’alembert’s test and Cauchy’s radical test are not independent one from another. It
is proposed a transition scheme from one test to another and vice versa. The equivalence of the tests is proved
for series with the monotonically decreasing terms. This fact is used to formulate a new test of the
convergence for series with positive terms. The test is equivalent to Dalembert and Cauchy’s radical tests, but
it has some advantages. It can be applied to any series within Cauchy-D’alembert’s theory.
Keywords: series, convergence, comparison tests, D’alembert’s test, Cauchy’s test, limit.
В статье показано, что признак Даламбера и радикальный признак Коши не являются независимыми
друг от друга. Предложена схема перехода от одного признака к другому. Эквивалентность признаков
доказана для рядов с монотонно убывающими членами. Этот факт использован для формулировки
нового признака сходимости рядов с положительными членами. Новый признак эквивалентен признакам
Даламбера и Коши, но имеет некоторые преимущества – он может применяться к любым рядам в
рамках теории Коши-Даламбера.
Ключевые слова: ряд, сходимость, признак сравнения, признак Даламбера, признак Коши, предел.
У статті показано, що ознака Даламбера і радикальна ознака Коші не є незалежними одна від другої.
Запропоновано схему переходу від однієї ознаки до другої. Еквівалентність ознак доведена для рядів з
монотонно регресними членами. Цей факт використан для формулювання нової ознаки збіжності рядів з
додатними членами. Нова ознака еквівалентна ознакам Даламбера і Коші, але має деяку перевагу вона
може застосовуватися до будь-яких рядів в рамках теорії Коші-Даламбера.
Ключові слова: ряд, збіжність, ознака порівняння, ознака Даламбера, ознака Коші, границя.
Introduction
Tests of the convergence of series with positive terms such as Cauchy’s and D’alembert’s
tests are usually applied to quickly converging series. In the limit form for any series∑
∞
=1n
n
u ,
0>
n
u these tests can be written down in the forms of inequalities [1]:
0 ,1lim ≥≤
∞→
n
n
n
n
uu , (1)
1lim
1
≤
+
∞→
n
n
n u
u , 0>
n
u . (2)
Mironenko L.P.
«Искусственный интеллект» 2013 № 3 508
6М
А
The first inequality is called Cauchy’s radical test, the second one is D’alembert’s test. For both
tests the sign of equality means uncertainty in a question of convergence of the series. In that
case additional research is required. They require of application of more " delicate " tests.
The proof of both tests (1) and (2) are based on the comparison test with respect to the geo-
metrical series∑
∞
=0n
n
q , 1<q . From this we can make a conclusion about equivalence of
D’alembert’s and Cauchy’s tests. In the literature the direct connection between these tests is
usually not considered, but only through the geometrical series (Figure.1).
Figure. 1 – The scheme of a connection between D’alembert’s and Cauchy’s tests
In this work we will show a direct connection between D’alembert’s and Cauchy’s tests,
moreover, how from one of the test follows another test and vice versa.
In the course of the proof of the tests there is appeared one more form of the test, which
is equivalent both to D’alembert’s and Cauchy’s tests. This test is no more than the refor-
mulation of D’alembert’s and Cauchy’s tests, but it has some advantages in comparison with
D’alembert’s and Cauchy’s tests.
D’alembert’s test as a corollary from Cauchy’s radical test
and vice versa
Let us address to Cauchy’s radical test (1), and we will execute some
transformations. We apply logarithm to both parts of the inequality (1) ( ) 1lnlimln ≤
∞→
n
n
n
u . It
is clear, if 0lim =
∞→
n
n
n
u the logarithm ( )ln lim n
n
n
u
→∞
does not exist. In that case the range of
definition must be extended to −∞=0ln and the inequality remains fair. This moment of
the proof can be avoided if to consider not the limit form of the test, but to consider the
inequality 1<n
n
u beginning from some number
o
N .
A property of continuity of the function xy ln= allows inserting the logarithm sign
after of the limit sign
0
ln
lim0lnlim ≤⇒≤
∞→∞→ n
u
u
n
n
n
n
n
.
According to the necessary test we have 0→
n
u . Then, −∞→
n
uln at ∞→n . The limit has
uncertainty
∞
∞
and the conditions of L’Hospital’s rule are satisfied if the terms of the se-
quence }{
n
u are monotonically decreasing. Therefore
( ) ( )
0lim
ln
lim ≤
′
=
′
′
∞→∞→
n
n
n
n
n u
u
n
u
.
The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory…
«Штучний інтелект» 2013 № 3 509
6М
Let's represent the derivative ( )
n
u
′
at ∞→n as
( )
nnn
uuu −=
′
+1
. (3)
The proof of this formula is based on mean value Lagrange’s theorem [2]. Thus we suppose
the sequence }{
n
u is monotonically decreasing at ∞→n . Only in this case the derivative is
existed.
After of the substitution of the expression (3) to the previous inequality we will receive:
.1lim0limlim0lim
111
≤⇒≤−⇒≤
−
+
∞→∞→
+
∞→
+
∞→
n
n
n
n
n
n
n
n
n
n
nn
n u
u
u
u
u
u
u
uu
It is possible the reverse solution of the problem. Starting from D’alembert’s test (2) it is
not difficult to get Cauchy’s test. We make a conclusion about of the equivalence of these tests.
In any of the cases of the proof of the equivalence of the tests we receive an interme-
diate product - the inequality ( ) 0lnlim ≤
′
∞→
n
n
u .
A new test of a convergence of series with positive terms
From the scheme of the proof of equivalence of D’alembert’s and Cauchy’s tests we have
an inequality which can be considered as a new test
( ) 0lnlim ≤
′
∞→
n
n
u , 0>
n
u . (4)
The sign ‘=’ in the condition (4) means uncertainty as well as in Cauchy-D’alembert
theory. The problem of a convergence needs in additional research, for example, to apply
any other test of convergence.
This remark in accuracy coincides with the cases in the tests (1) and (2).
The proof of the test (4) can be done independently by using geometrical series
∑
∞
=0n
n
q , 1<q . Compare the given series∑
∞
=0n
n
u , 0>
n
u with the geometrical series
q
n
u
qnququ
nn
n
n
n
ln
ln
lnlnln <⇒=<⇒< .
The right-hand side of the inequality is negative and in the limit at ∞→n the rigorous
sign in the inequality must be changed on non-rigorous one:
0
ln
limln
ln
lim ≤⇒≤
∞→∞→ n
u
q
n
u
n
n
n
n
If we apply L’Hospital’s rule to the left-hand side of the inequality, we get the test (4).
The test (4) contains derivative, therefore we can call the test as the differential form
of D’alembert-Cauchy’s tests, or shortly, the differential test. This test can be applied to any
series. Therefore we will consider some examples.
Example 1. To investigate the convergence of the series∑
∞
= +
++
+
1 23
532
2
n n
nn
n
.
Here, ( ) ( ) ( ),53ln
2
1
2ln32lnln
2
++−+−+= nnnnu
n
( )
( )
,
532
32
2ln
2
1
ln
2
++
+
−−
+
=
′
nn
n
n
u
n
( ) .02lnlnlim <−=
′
∞→
n
n
u The series is convergent.
Example 2. To investigate the convergence of the series∑
∞
=
++
+
1 3 47
53
2
n
nn
n .
Mironenko L.P.
«Искусственный интеллект» 2013 № 3 510
6М
А
Here, ( ) ( ),53ln
3
1
2lnln
47
++−+= nnnu
n
( )
( )
,
533
127
2
1
ln
47
36
++
+
−
+
=
′
nn
nn
n
u
n
( ) .0lnlim =
′
∞→
n
n
u The test does not work.
Example 3. To investigate the convergence of the series ∑
∞
=
−
1
!
n
nn
n
en
.
Here, ,!lnlnlnln nennnu
n
−−=
( )
( )
,0
!
!
11lnln =
′
−−+=
′
n
n
nu
n
( ) .0lnlim =
′
∞→
n
n
u The test does not work.
Here the asymptotic formula ( ( ) nnn ln!! =
′
is used (APPENDIX).
The first example shows an application of our test, the second shows, when the test has
the same restrictions that D’alembert’s and Cauchy’s tests. The third example shows possibility
of application of the test to series, which contain the factorial function !n .
Conclusions
In the paper two main results are received. First, the equivalence of D’alembert’s and
Cauchy’s tests is proved for monotonically series. It is shown, how from one of the test follows
the other.
In the process of the proof it has formulated a new test which is equivalent to
D’alembert’s and Cauchy’s tests. We also have shown how the differential test arises from
the comparison test with respect to the geometrical series.
Now the scheme that was represented in the Fig. 1 becomes as it shown in the Figure. 2.
Figure. 2. – The scheme demonstrates three equivalent tests
As a rule, D’alembert’s test is applied when series’ terms contain the factorial !n . Cauchy’s
test does not work in this case. The operation n n! is not defined. From this point of view
D’alembert’s and Cauchy’s tests supplement each other. Our test can be applied in both si-
tuations as in the case !n as in any cases as well. Our differential test is more universal than
D’alembert’s and Cauchy’s tests, but it has a very important restriction. It can be applied
only to series with monotonically decreasing terms.
In the case of !n we propose to use the approximate formula ( ) nnn
n
ln!!
1>>
≈
′
. This formula
allows applying our test to such series for which there is the problem of a choice what test
should be used: D’alembert’s or Cauchy’s test.
The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory…
«Штучний інтелект» 2013 № 3 511
6М
Application of our test is not more difficult than D’alembert’s and Cauchy’s tests and
in many cases it is easier them. Once more we will emphasize that our test has no alternative
in the case of application, because logarithm of a product or division of functions is equal to their
sum and difference. Derivative of such objects is calculated simply even it is easier than for pro-
duct and division.
APPENDIX. The derivative of the factorial function !n .
Let's begin from Stirling’s formula
++++= −−
)(
51840
139
288
1
12
1
12!
4
32
no
nnn
ennn
nn
π ,
We will restrict ourselves only with of two first terms of the sum. We take logarithm
of the equality
.
12
1
1lnlnln2ln!ln
++++= −
n
ennu
nn
π
Let's accept approximation ( ) ,12/112/11ln nn ≈+ then
,
12
1
lnln
2
1
2ln!ln
n
nnnnu +−++≈ π
( )
2
ln ! 1 1
ln .
! 2 12
n
n
n n n
′
≈ + −
Then it follows that ( ) nn
nn
nnn
n
ln!
12
1
2
1
ln!!
1
2
>>
≈
−+≈
′
.
Literature
1. Kudryavtsev L.D. Matematicheski analiz. - Tom I., Nauka, 1970 - 571 s.
2. Wrede R.C., Spiegel M. Theory and problems of advanced calculus 2002, 433s
3. Fichtengoltz G.M. Kurs differentcialnogo i integralnogo ischislenia, Tom. 2, Nauka, «FizML», 1972 - 795 s.
4. . Apostol T.M. Calculus. One-Variable Calculus with an Introduction to Linear Algebra. Vol 1. – John
Wilay and Sons, Inc., 1966, 667 with.
5. .Ilyin V. A, Pozdnyak E.G. Osnovi matematicheskiogo analiza. – Tom 1, FMF, Moskva, 1956. - 472s
RESUME
L.P.Mironenko
The proof of an equivalence of D’alembert’s and Cauchy’s tests in
the theory of numerical series
In the paper it is shown that D’alembert’s test and Cauchy’s radical test are not
independent one from another. It is proposed a transition scheme from one test to another
and vice versa. The equivalence of the tests is proved for series with the monotonically
decreasing terms. This fact is used to formulate a new test of the convergence for series
with positive terms. The test is equivalent to Dalembert and Cauchy’s radical tests, but it
has some advantages. It can be applied to any series within Cauchy-D’alembert’s theory.
Статья поступила в редакцию 26.04.2013
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