Abel–Dini–Pringsheim theorem

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.  
  2. For all   we have  
  3. 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.