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Put the historical notes in the end again. Most (if not all) of the math textbooks that I see that have historical notes section have them at the end of each chapter, whether it's short or not. I believe the readers can handle that!
 
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In [[calculus]], the '''Abel–Dini–Pringsheim theorem''' is a [[convergence test]] which constructs from a [[divergent series]] a series that diverges more slowly, and from [[convergent series]] one that converges more slowly.<ref name="Knopp">{{cite book |last=Knopp |first=Konrad |url=https://archive.org/details/theoryandapplica031692mbp/ |title=Theory and application of infinite series |date=1951 |publisher=Blackie & Son |others=Translated from the 2nd edition and revised in accordance with the fourth by R. C. H. Young. |edition=2 |location=London–Glasgow |language=en |translator-last=Young |translator-first=R. C. H. |zbl=0042.29203}}</ref>{{rp|§IX.39}} Consequently, for every convergence test based on a particular series there is a series about which the test is inconclusive.<ref name="Knopp" />{{Rp|page=299}} For example, the [[Raabe test]] is essentially a comparison test based on the family of series whose <math>n</math>th term is <math>1/n^t</math> (with <math>t\in\mathbb R</math>) and is therefore inconclusive about the series of terms <math>1/(n\ln n)</math> which diverges more slowly than the [[harmonic series (mathematics)|harmonic series]].
In [[calculus]], the '''Abel–Dini–Pringsheim theorem''' is a [[convergence test]] which constructs from a [[divergent series]] a series that diverges more slowly, or similarly for [[convergent series]].<ref name="Knopp">{{cite book
|last=Knopp
|first=Konrad
|translator-last=Young
|translator-first=R. C. H.
|title=Theory and application of infinite series
|language=en
|edition=2
|others=Translated from the 2nd edition and revised in accordance with the fourth by R. C. H. Young.
|publisher=Blackie & Son
|location=London–Glasgow
|date=1951
|zbl=0042.29203
}}</ref>{{rp|§IX.39}} Consequently, for every convergence test based on a particular series there is a series about which the test is inconclusive.


== Definitions ==
== Definitions ==
The Abel–Dini–Pringsheim theorem can be given for divergent series or convergent series. Helpfully, these definitions are equivalent, and it suffices to prove only one case. This is because applying the Abel–Dini–Pringsheim theorem for divergent series to the series with partial sum
:<math>S_n'=\frac1{r_n}</math>
yields the Abel–Dini–Pringsheim theorem for convergent series.<ref name="Hildebrandt">{{cite journal |last=Hildebrandt |first=T. H. |date=1942 |title=Remarks on the Abel-Dini theorem |journal=American Mathematical Monthly |language=en |volume=49 |issue=7 |pages=441–445 |doi=10.2307/2303268 |issn=0002-9890 |jstor=2303268 |mr=0007058 |zbl=0060.15508}}</ref>

=== For divergent series ===
=== For divergent series ===
Suppose that <math>(a_n)_{n=0}^\infty\subset(0,\infty)</math> is a sequence of [[positive real number]]s such that the series
Suppose that <math>(a_n)_{n=0}^\infty\subset(0,\infty)</math> is a sequence of [[positive real number]]s such that the series
:<math>\sum_{n=0}^\infty a_n=\infty</math>
:<math>\sum_{n=0}^\infty a_n=\infty</math>
diverges to infinity. Let <math>S_n=a_0+a_1+\cdots+a_n</math> denote the <math>n</math>th [[partial sum]]. The '''Abel–Dini–Pringsheim theorem''' for divergent series states that the following conditions hold.
diverges to infinity. Let <math>S_n=a_0+a_1+\cdots+a_n</math> denote the <math>n</math>th [[partial sum]]. The '''Abel–Dini–Pringsheim theorem''' for divergent series states that the following conditions hold.

* <math>\sum_{n=0}^\infty\frac{a_n}{S_n}=\infty</math>
* For all <math>\epsilon>0</math> we have <math>\sum_{n=1}^\infty\frac{a_n}{S_nS_{n-1}^\epsilon}<\infty</math>
# <math>\sum_{n=0}^\infty\frac{a_n}{S_n}=\infty</math>
# For all <math>\epsilon>0</math> we have <math>\sum_{n=1}^\infty\frac{a_n}{S_nS_{n-1}^\epsilon}<\infty</math>
* If also <math>\lim_{n\to\infty}\frac{a_n}{S_n}=0</math>, then <math>\lim_{n\to\infty}\frac{a_0/S_0+a_1/S_1+\cdots+a_n/S_n}{\ln S_n}=1</math>
# If also <math>\lim_{n\to\infty}\frac{a_n}{S_n}=0</math>, then <math>\lim_{n\to\infty}\frac{a_0/S_0+a_1/S_1+\cdots+a_n/S_n}{\ln S_n}=1</math>
Consequently, the series
Consequently, the series
:<math>\sum_{n=0}^\infty\frac{a_n}{S_n^t}</math>
:<math>\sum_{n=0}^\infty\frac{a_n}{S_n^t}</math>
converges if <math>t>1</math> and diverges if <math>t\le1</math>.
converges if <math>t>1</math> and diverges if <math>t\le1</math>. When <math>t\le1</math>, this series diverges less rapidy than <math>a_n</math>.<ref name="Knopp" />
{{math proof|proof=
{{math proof|proof=
'''Proof of the first part.''' By the assumption <math>S_n</math> is nondecreasing and diverges to infinity. So, for all <math>n\in\{0,1,2,\dots\}</math> there is <math>k_n\in\{0,1,2,\dots\}</math> such that
'''Proof of the first part.''' By the assumption <math>S_n</math> is nondecreasing and diverges to infinity. So, for all <math>n\in\{0,1,2,\dots\}</math> there is <math>k_n\geq 1</math> such that
:<math>\frac{S_n}{S_{n+k_n}}<\frac12</math>
:<math>\frac{S_n}{S_{n+k_n}}<\frac12</math>
Therefore
Therefore
:<math>\frac{a_n}{S_n}+\cdots+\frac{a_{n+k_n}}{S_{n+k_n}}\ge\frac{a_n+\cdots+a_{n+k_n}}{S_{n+k_n}}=\frac{S_{n+k_n}-S_{n}}{S_{n+k_n}}=1-\frac{S_n}{S_{n+k_n}}>\frac12</math>
:<math>\frac{a_{n+1}}{S_{n+1}}+\cdots+\frac{a_{n+k_n}}{S_{n+k_n}}\ge\frac{a_{n+1}+\cdots+a_{n+k_n}}{S_{n+k_n}}=\frac{S_{n+k_n}-S_{n}}{S_{n+k_n}}=1-\frac{S_n}{S_{n+k_n}}>\frac12</math>
and hence <math>a_0/S_0+\cdots+a_n/S_n</math> is not a [[Cauchy sequence]]. This implies that the series
and hence <math>a_0/S_0+\cdots+a_n/S_n</math> is not a [[Cauchy sequence]]. This implies that the series
:<math>\sum_{n=0}^\infty\frac{a_n}{S_n}</math>
:<math>\sum_{n=0}^\infty\frac{a_n}{S_n}</math>
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:<math>f'(x)=\epsilon(x^{\epsilon-1}-1)\ge0\qquad(\forall x\in(0,1])</math>
:<math>f'(x)=\epsilon(x^{\epsilon-1}-1)\ge0\qquad(\forall x\in(0,1])</math>
:<math>f'(x)=\epsilon(x^{\epsilon-1}-1)\le0\qquad(\forall x\in[1,\infty)).</math>
:<math>f'(x)=\epsilon(x^{\epsilon-1}-1)\le0\qquad(\forall x\in[1,\infty)).</math>
(Alternatively, <math>g(x)=1-x^\epsilon</math> is convex and its tangent at <math>1</math> is <math>y=g'(1)(x-1)=\epsilon(1-x)</math>)
Therefore,
Therefore,
:<math>\begin{align}
:<math>\begin{align}
Line 76: Line 69:
In particular, the series
In particular, the series
:<math>\sum_{n=0}^\infty\frac{a_n}{r_n^t}</math>
:<math>\sum_{n=0}^\infty\frac{a_n}{r_n^t}</math>
is convergent when <math>t<1</math>, and divergent when <math>t\ge1</math>.
is convergent when <math>t<1</math>, and divergent when <math>t\ge1</math>. When <math>t<1</math>, this series converges more slowly than <math>a_n</math>.<ref name="Knopp" />

=== Equivalence ===
Applying the Abel–Dini–Pringsheim theorem for divergent series to the series with partial sum
:<math>S_n'=\frac1{r_n}</math>
yields the Abel–Dini–Pringsheim theorem for convergent series.<ref name="Hildebrandt">{{cite journal
|last=Hildebrandt
|first=T. H.
|title=Remarks on the Abel-Dini theorem
|language=en
|journal=American Mathematical Monthly
|volume=49
|pages=441–445
|date=1942
|issue=7
|issn=0002-9890
|doi=10.2307/2303268
|jstor=2303268
|mr=0007058
|zbl=0060.15508
}}</ref> Therefore, the two forms of the theorems are in fact equivalent.


== Examples ==
== Examples ==
Line 113: Line 86:
:<math>\lim_{n\to\infty}\frac{1+1/(2\ln 2)+\cdots+1/(n\ln n)}{\ln\ln n}=1.</math>
:<math>\lim_{n\to\infty}\frac{1+1/(2\ln 2)+\cdots+1/(n\ln n)}{\ln\ln n}=1.</math>


== History ==
== Historical notes ==
[[Niels Henrik Abel]] proved a weak form of the first part of the theorem (for divergent series).<ref name="AbelNHNoteSurLeMémoire">{{cite journal |date=1828 |doi=10.1515/crll.1828.3.79 |first=Niels Henrik |issn=0075-4102 |journal=Journal für die Reine und Angewandte Mathematik |language=fr |last=Abel |mr=1577677 |pages=79–82 |title=Note sur le mémoire de Mr. L. Olivier No. 4. du second tome de ce journal, ayant pour titre "remarques sur les séries infinies et leur convergence." Suivi d'une remarque de Mr. L. Olivier sur le même objet |url=https://eudml.org/doc/183130 |volume=3 |zbl=003.0093cj}}</ref> [[Ulisse Dini]] proved the complete form and a weak form of the second part.<ref name="DiniSulleSerie">{{cite journal
The theorem was proved in three parts. [[Niels Henrik Abel]] proved a weak form of the first part of the theorem (for divergent series).<ref name="AbelNHNoteSurLeMémoire">{{cite journal |last=Abel |first=Niels Henrik |date=1828 |title=Note sur le mémoire de Mr. L. Olivier No. 4. du second tome de ce journal, ayant pour titre "remarques sur les séries infinies et leur convergence." Suivi d'une remarque de Mr. L. Olivier sur le même objet |url=https://eudml.org/doc/183130 |journal=Journal für die Reine und Angewandte Mathematik |language=fr |volume=3 |pages=79–82 |doi=10.1515/crll.1828.3.79 |issn=0075-4102 |mr=1577677}}</ref> [[Ulisse Dini]] proved the complete form and a weak form of the second part.<ref name="DiniSulleSerie">{{cite journal |last1=Dini |first1=Ulisse |date=1868 |title=Sulle serie a termini positivi |journal=Giornale di Matematiche |language=it |volume=6 |pages=166–175 |jfm=01.0082.01}}</ref> [[Alfred Pringsheim]] proved the second part of the theorem.<ref name="Pringsheim">{{cite journal |last1=Pringsheim |first1=Alfred |date=1890 |title=Allgemeine Theorie der Divergenz und Convergenz von Reihen mit positiven Gliedern |url=https://eudml.org/doc/157469 |journal=Mathematische Annalen |language=de |volume=35 |issue=3 |pages=297–394 |doi=10.1007/BF01443860 |issn=0025-5831 |jfm=21.0230.01}}</ref> The third part is due to [[Ernesto Cesàro]].<ref name="CesàroENouvellesRemarques">{{cite journal |last1=Cesàro |first1=Ernesto |date=1890 |title=Nouvelles remarques sur divers articles concernant la théorie des séries |url=http://www.numdam.org/item/?id=NAM_1890_3_9__353_0 |journal=Nouvelles annales de mathématiques: Journal des candidats aux écoles polytechnique et normale, Serie 3 |language=fr |volume=9 |pages=353–367 |issn=1764-7908 |jfm=22.0247.02}}</ref>
|last1=Dini
|first1=Ulisse
|title=Sulle serie a termini positivi
|language=it
|journal=Giornale di Matematiche
|volume=6
|pages=166–175
|date=1868
|jfm=01.0082.01
}}</ref> [[Alfred Pringsheim]] proved the second part of the theorem.<ref name="Pringsheim">{{cite journal |date=1890 |doi=10.1007/BF01443860 |first1=Alfred |issn=0025-5831 |jfm=21.0230.01 |journal=Mathematische Annalen |language=de |last1=Pringsheim |pages=297–394 |title=Allgemeine Theorie der Divergenz und Convergenz von Reihen mit positiven Gliedern |url=https://eudml.org/doc/157469 |volume=35|issue=3 }}</ref> The third part is due to [[Ernesto Cesàro]].<ref name="CesàroENouvellesRemarques">{{cite journal |date=1890 |first1=Ernesto |issn=1764-7908 |jfm=22.0247.02 |journal=Nouvelles annales de mathématiques: Journal des candidats aux écoles polytechnique et normale, Serie 3 |language=fr |last1=Cesàro |pages=353–367 |title=Nouvelles remarques sur divers articles concernant la théorie des séries |url=http://www.numdam.org/item/?id=NAM_1890_3_9__353_0 |volume=9}}</ref>


== References ==
== References ==
{{reflist}}
{{reflist}}


{{DEFAULTSORT:Abel-Dini-Pringsheim theorem}}
[[Category:Convergence tests]]
[[Category:Convergence tests]]

Latest revision as of 15:21, 1 July 2024

In calculus, the Abel–Dini–Pringsheim theorem is a convergence test which constructs from a divergent series a series that diverges more slowly, and from convergent series one that converges more slowly.[1]: §IX.39  Consequently, for every convergence test based on a particular series there is a series about which the test is inconclusive.[1]: 299  For example, the Raabe test is essentially a comparison test based on the family of series whose th term is (with ) and is therefore inconclusive about the series of terms which diverges more slowly than the harmonic series.

Definitions

[edit]

The Abel–Dini–Pringsheim theorem can be given for divergent series or convergent series. Helpfully, these definitions are equivalent, and it suffices to prove only one case. This is because applying the Abel–Dini–Pringsheim theorem for divergent series to the series with partial sum

yields the Abel–Dini–Pringsheim theorem for convergent series.[2]

For divergent series

[edit]

Suppose that is a sequence of positive real numbers such that the series

diverges to infinity. Let denote the th partial sum. The Abel–Dini–Pringsheim theorem for divergent series states that the following conditions hold.

  1. For all we have
  2. If also , then

Consequently, the series

converges if and diverges if . When , this series diverges less rapidy than .[1]

Proof

Proof of the first part. By the assumption is nondecreasing and diverges to infinity. So, for all there is such that

Therefore

and hence is not a Cauchy sequence. This implies that the series

is divergent.

Proof of the second part. If , we have for sufficiently large and thus . So, it suffices to consider the case . For all we have the inequality

This is because, letting

we have

(Alternatively, is convex and its tangent at is ) Therefore,

Proof of the third part. The sequence is nondecreasing and diverges to infinity. By the Stolz-Cesaro theorem,

For convergent series

[edit]

Suppose that is a sequence of positive real numbers such that the series

converges to a finite number. Let denote the th remainder of the series. According to the Abel–Dini–Pringsheim theorem for convergent series, the following conditions hold.

  • For all we have
  • If also then

In particular, the series

is convergent when , and divergent when . When , this series converges more slowly than .[1]

Examples

[edit]

The series

is divergent with the th partial sum being . By the Abel–Dini–Pringsheim theorem, the series

converges when and diverges when . Since converges to 0, we have the asymptotic approximation

Now, consider the divergent series

thus found. Apply the Abel–Dini–Pringsheim theorem but with partial sum replaced by asymptotically equivalent sequence . (It is not hard to verify that this can always be done.) Then we may conclude that the series

converges when and diverges when . Since converges to 0, we have

Historical notes

[edit]

The theorem was proved in three parts. Niels Henrik Abel proved a weak form of the first part of the theorem (for divergent series).[3] Ulisse Dini proved the complete form and a weak form of the second part.[4] Alfred Pringsheim proved the second part of the theorem.[5] The third part is due to Ernesto Cesàro.[6]

References

[edit]
  1. ^ a b c d Knopp, Konrad (1951). Theory and application of infinite series. Translated by Young, R. C. H. Translated from the 2nd edition and revised in accordance with the fourth by R. C. H. Young. (2 ed.). London–Glasgow: Blackie & Son. Zbl 0042.29203.
  2. ^ Hildebrandt, T. H. (1942). "Remarks on the Abel-Dini theorem". American Mathematical Monthly. 49 (7): 441–445. doi:10.2307/2303268. ISSN 0002-9890. JSTOR 2303268. MR 0007058. Zbl 0060.15508.
  3. ^ Abel, Niels Henrik (1828). "Note sur le mémoire de Mr. L. Olivier No. 4. du second tome de ce journal, ayant pour titre "remarques sur les séries infinies et leur convergence." Suivi d'une remarque de Mr. L. Olivier sur le même objet". Journal für die Reine und Angewandte Mathematik (in French). 3: 79–82. doi:10.1515/crll.1828.3.79. ISSN 0075-4102. MR 1577677.
  4. ^ Dini, Ulisse (1868). "Sulle serie a termini positivi". Giornale di Matematiche (in Italian). 6: 166–175. JFM 01.0082.01.
  5. ^ Pringsheim, Alfred (1890). "Allgemeine Theorie der Divergenz und Convergenz von Reihen mit positiven Gliedern". Mathematische Annalen (in German). 35 (3): 297–394. doi:10.1007/BF01443860. ISSN 0025-5831. JFM 21.0230.01.
  6. ^ Cesàro, Ernesto (1890). "Nouvelles remarques sur divers articles concernant la théorie des séries". Nouvelles annales de mathématiques: Journal des candidats aux écoles polytechnique et normale, Serie 3 (in French). 9: 353–367. ISSN 1764-7908. JFM 22.0247.02.