Download e-book for kindle: A Course of Mathematical Analysis (Vol. 2) by Nikolsky S.M.

By Nikolsky S.M.

A textbook for college scholars (physicists and mathematicians) with designated supplementary fabric on mathematical physics. in accordance with the path learn through the writer on the Moscow Engineering Physics Institute. quantity 2 includes a number of integrals, box idea, Fourier sequence and Fourier necessary, differential manifolds and differential types, and the Lebesgue vital.

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0 dagegen sind die Näherungsbrüche von +~ + . I... bzw. I + ... [1 kleiner als die der ersten Ordnung, also kleiner als a;' b 1 foour az > 0 b zw. b _a;' 1 foour a 2 1 < 0, nach unsern Voraussetzungen also beidemal < 1. Daher sind die Näherungsbrüche des extendierten Kettenbruches> bo - 1, so daß für den Hälifungswert ~o die Ungleichung gilt: (b) ~o > bo -1, falls a 1 < O. § 13. Irrationalität gewisser Kettenbrüche Wäre nun im Fall (a) etwa ;0 = bo + 55 bo, also all + jb; a 2 + ... = 1 ~ b0' + I~:I + la~1 = bo nach Satz 4 des § 1 auch so hätte man wegen der Identität bo °= b2 + a3 Ib3 + 1 a4 Ib4 + ....

K 2 4

II. 10 ist vielfach etwas unbequem. Man kann aber daraus bequemere Bedingungen herleiten, welche für die Konvergenz wenigstens hinreichen. Sind nämlich u v ' Vv reelle Zahlen > 0, so ist 2 ruvv v < Uv + Vv' Wenn also die Reihe L Vuvvv divergiert, so wird mindestens eine der beiden Reihen ebenfalls divergieren müssen. 10 vorkommenden Reihen in der Weise an, daß man zuerst LUv, LVv sodann auch Uv = a l aa ... a 2v- 1 b2V ' Vv = a 2 a4. • ... a 2va 2v- 2 b2V - 1 ' also u v = setzt, so ergibt sich, daß sowohl die Divergenz der Reihe der Reihe kommt b 2v v 1 b 2v _ 1 b 2v 'V a 2V L Vb2vb2v+1 , als auch a2V+l für die Konvergenz des Kettenbruches ausreicht.

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