Sapogov's criterion

The Sapogow criterion is one of the convergence criteria for infinite series and as such belongs to the mathematical branch of analysis . As GM Fichtenholz shows in Volume II of his three-volume differential and integral calculus , it goes back to the Soviet mathematician Nikolai Alexandrowitsch Sapogow (1915–1983).

formulation

Following spruce wood, the criterion can be formulated as follows:

A monotonically increasing sequence of positive real numbers is given . ${\displaystyle {\mathcal {A}}=\left(a_{n}\right)_{n=1,2,3,\ldots }}$

This is the turn

${\displaystyle S=\sum _{n=1}^{\infty }{\left(1-{\frac {a_{n}}{a_{n+1}}}\right)}}$

educated. Then:

(I) is a convergent series if is a bounded sequence . In this case the related series is also convergent. ${\displaystyle S}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \textstyle S^{*}=\sum _{n=1}^{\infty }{\left({\frac {a_{n+1}}{a_{n}}}-1\right)}}$

(II) However, if it is unbounded, then it is divergent. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle S}$

Related criteria

Connected to the Sapogow criterion is another which goes back to Niels Henrik Abel and Ulisse Dini and with whose help Fichtenholz proves the Sapogow criterion. This criterion also appears in Konrad Knopp's monograph Theory and Application of Infinite Series , where it is referred to as Abel and Dini's theorem. Following the representation by Knopp, it can be specified as follows:

A sequence of positive real numbers and any real number are given . Let the series belonging to the sequence be divergent. ${\displaystyle \left(d_{n}\right)_{n=1,2,3,\ldots }}$ ${\displaystyle \alpha }$ ${\displaystyle \sum _{n=1}^{\infty }{d_{n}}}$

Then applies with regard to the partial sums sequence : ${\displaystyle \left(D_{n}\right)_{n=1,2,3,\ldots }=\left(d_{1},d_{1}+d_{2},d_{1}+d_{2}+d_{3},\ldots \right)}$

(a) For the newly formed series is also divergent. ${\displaystyle \alpha \leq 1}$ ${\displaystyle \sum _{n=1}^{\infty }{\frac {d_{n}}{{D_{n}}^{\alpha }}}}$

(b) For, however, is convergent. ${\displaystyle \alpha >1}$ ${\displaystyle \sum _{n=1}^{\infty }{\frac {d_{n}}{{D_{n}}^{\alpha }}}}$

Knopp traces Abel and Dini's theorem to a result which comes from Alfred Pringsheim and which Knopp calls Pringsheim's theorem :

If a sequence of positive real numbers with partial sum sequence and the series belonging to the sequence is divergent, then for any real number is the related series ${\displaystyle \left(d_{n}\right)_{n=1,2,3,\ldots }}$ ${\displaystyle \left(D_{n}\right)_{n=1,2,3,\ldots }}$ ${\displaystyle \sum _{n=1}^{\infty }{d_{n}}}$ ${\displaystyle \beta >0}$

${\displaystyle \sum _{m=2}^{\infty }{\frac {d_{m}}{D_{m}\cdot {D_{m-1}}^{\beta }}}}$

always convergent.

literature

• NH Abel: 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” . In: Journal for pure and applied mathematics . tape 3 , 1828, p. 79-82 ( MR1577677 ).
• U. Dini: Sulle serie a termini positivi . In: Annali delle Università Toscane . tape 9 , 1867, p. 41–76 ( serie a termini positivi (Wikisource) ).
• GM Fichtenholz: Differential and Integral Calculus II . Translation from Russian and scientific editing: Dipl.-Math. Brigitte Mai, Dipl.-Math. Walter Mai (=  university books for mathematics . Volume 62 ). 6th edition. VEB Deutscher Verlag der Wissenschaften , Berlin 1974.
• Konrad Knopp: Theory and Application of the Infinite Series (=  The Basic Teachings of Mathematical Sciences . Volume 2 ). 5th, corrected edition. Springer Verlag , Berlin, Göttingen, Heidelberg, New York 1964, ISBN 3-540-03138-3 ( MR0183997 ).
• Alfred Pringsheim: General theory of the divergence and convergence of series with positive terms . In: Mathematical Annals . tape 35 , 1890, pp. 297-394 .

Individual evidence

1. ↑ GM Fichtenholz: differential and integral calculus II. 1974, p. 304, p. 834
2. ↑ Although the Soviet Union did not yet exist in the year of Sapogov's birth, Fichtenholz calls Sapogow a “Soviet mathematician”.
3. ↑ Fichtenholz, op.cit., P. 304
4. ↑ Fichtenholz, op. Cit., Pp. 303-304
5. ↑ Konrad Knopp: Theory and application of the infinite series. 1964, p. 299
6. ↑ Knopp, op.cit., P. 300