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Dirixle alomati (testi)

Matematikada Dirixlet alomati (testi) qatorlarning yaqinlashuvchiligini tekshirish usuli hisoblanadi. U oʻz muallifi Peter Gustav Lejeune Dirixle sharafiga nomlangan va 1862-yilda Journal de Mathématiques Pures et Appliquées jurnalida muallifning vafotidan keyin nashr etilgan.

Sharti

Agar {an}{\displaystyle \{a_n\}} haqiqiy sonlar ketma- ketligi va {bn}{\displaystyle \{b_n\}} kompleks sonlar ketma-ketligi

  • {an}{\displaystyle \{a_n\}} monoton o'smovchi (monoton o'smaydigan)
  • limn→∞⁡an=0{\displaystyle \lim_{n \to \infty}a_n = 0}
  • |∑n=1Nbn|≤M{\displaystyle \left|\sum^{N}_{n=1}b_n\right|\leq M}, barcha musbat butun N uchun

shartlarni qanoatlantirsa, u holda∑n=1∞anbn{\displaystyle \sum_{n=1}^\infty a_{n}b_{n}}qator yaqinlashadi, bu yerda M qandaydir konstanta.

Isbot

Sn=∑k=1nakbk{\displaystyle S_n = \sum_{k=1}^n a_k b_k} va Bn=∑k=1nbk{\displaystyle B_n = \sum_{k=1}^n b_k} bo'lsin.

Qismlar bo'yicha yig'ish natijasida Sn=anBn+∑k=1n−1Bk(ak−ak+1){\displaystyle S_n = a_{n} B_{n} + \sum_{k=1}^{n-1} B_k (a_k - a_{k+1})} ni hosil qilamiz. Bn{\displaystyle B_n} ning M bilan chegaralanganligi va an→0{\displaystyle a_n \to 0} bo'lganligi uchun, bu hadlarning birinchisi n→∞{\displaystyle n\to\infty} da nolga yaqinlashadi, anBn→0{\displaystyle a_{n} B_{n} \to 0}.

Har bir k uchun, |Bk(ak−ak+1)|≤M|ak−ak+1|{\displaystyle |B_k (a_k - a_{k+1})| \leq M|a_k - a_{k+1}|} munosabat o'rinli. Ammo, agar {an}{\displaystyle \{a_n\}} kamayuvchi bo'lsa,∑k=1nM|ak−ak+1|=∑k=1nM(ak−ak+1)=M∑k=1n(ak−ak+1),{\displaystyle \sum_{k=1}^{n} M|a_k - a_{k+1}| = \sum_{k=1}^n M(a_k - a_{k+1}) = M\sum_{k=1}^n (a_k - a_{k+1}),}teleskopik yig'indi M(a1−an+1){\displaystyle M(a_1 - a_{n+1})} ga teng bo'ladi va shuning uchun n→∞{\displaystyle n \to \infty} da Ma1{\displaystyle Ma_1} ga yaqinlashadi. Shunday qilib, ∑k=1∞M(ak−ak+1){\displaystyle \sum_{k=1}^\infty M(a_k - a_{k+1})} yaqinlashuvchi bo'ladi. Shuningdek, agar {an}{\displaystyle \{a_n\}} o'suvchi bo'lsa,∑k=1nM|ak−ak+1|=−∑k=1nM(ak−ak+1)=−M∑k=1n(ak−ak+1),{\displaystyle \sum_{k=1}^{n} M|a_k - a_{k+1}| = -\sum_{k=1}^n M(a_k - a_{k+1}) = -M\sum_{k=1}^n (a_k - a_{k+1}),}teleskopik yig'indi −M(a1−an+1){\displaystyle -M(a_1 - a_{n+1})} ga teng bo'lib, n→∞{\displaystyle n\to\infty} da −Ma1{\displaystyle -Ma_1} ga yaqinlashadi. Shunday qilib, yana, ∑k=1∞M(ak−ak+1){\displaystyle \sum_{k=1}^\infty M(a_k - a_{k+1})} yaqinlashuvchi bo'ladi.

Natijada, ∑k=1∞|Bk(ak−ak+1)|{\displaystyle \sum_{k=1}^\infty |B_k(a_k - a_{k+1})|} to'g'ridan- to'g'ri taqqoslash alomati natijasida yaqinlashuvchi bo'ladi. ∑k=1∞Bk(ak−ak+1){\displaystyle \sum_{k=1}^\infty B_k(a_k - a_{k+1})} qator esa absolyut yaqinlashish alomati natijasida yaqinlashadi. Shuning uchun Sn{\displaystyle S_n} yaqinlashuvchi bo'ladi.

Ilovalar

Dirixle alomatining xususiy holatlaridan biri bu amalda ko'proq qo'llaniladigan o'zgaruvchan qatorlarning yaqinlashish alomatidirbn=(−1)n⟹|∑n=1Nbn|≤1.{\displaystyle b_{n} = (-1)^n \Longrightarrow\left|\sum_{n=1}^N b_n\right| \leq 1.}Yana bir natija shuki, agar {an}{\displaystyle \{a_n\}} nolga yaqinlashadigan kamayuvchi ketma-ketlik bo'lsa, u holda ∑n=1∞ansin⁡n{\displaystyle \sum_{n=1}^\infty a_n \sin n} qator yaqinlashuvchi bo'ladi.

Xosmas integrallar

Xosmas integrallarning o'xshash yaqinlashuvchilik sharti bo'laklab integrallash yordamida isbotlangan. Agar f funksiyaning integrali barcha oraliqlarda tekis chegaralangan bo'lsa va g monoton kamayuvchi manfiy bo'lmagan funksiya bo'lsa, fg ning integrali yaqinlashuvchi xosmas integral bo'ladi.

Eslatmalar


Manbalar

  • Hardy, G. H., A Course of Pure Mathematics, Ninth edition, Cambridge University Press, 1946. (pp. 379–380).
  • Voxman, William L., Advanced Calculus: An Introduction to Modern Analysis, Marcel Dekker, Inc., New York, 1981. (§8.B.13–15) ISBN 0-8247-6949-X.

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