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
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Agar {\displaystyle \{a\_{n}\}} haqiqiy sonlar ketma- ketligi va {\displaystyle \{b\_{n}\}} kompleks sonlar ketma-ketligi

* {\displaystyle \{a\_{n}\}} monoton o'smovchi (monoton o'smaydigan)
* {\displaystyle \lim \_{n\to \infty }a\_{n}=0}
* {\displaystyle \left|\sum \_{n=1}^{N}b\_{n}\right|\leq M}, barcha musbat butun *N* uchun

shartlarni qanoatlantirsa, u holda



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{\displaystyle \sum \_{n=1}^{\infty }a\_{n}b\_{n}}

{\displaystyle \sum _{n=1}^{\infty }a_{n}b_{n}}qator yaqinlashadi, bu yerda *M* qandaydir konstanta.

Isbot
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{\textstyle S\_{n}=\sum \_{k=1}^{n}a\_{k}b\_{k}} va {\textstyle B\_{n}=\sum \_{k=1}^{n}b\_{k}} bo'lsin.

Qismlar bo'yicha yig'ish natijasida {\textstyle S\_{n}=a\_{n}B\_{n}+\sum \_{k=1}^{n-1}B\_{k}(a\_{k}-a\_{k+1})} ni hosil qilamiz. {\displaystyle B\_{n}} ning *M* bilan chegaralanganligi va {\displaystyle a\_{n}\to 0} bo'lganligi uchun, bu hadlarning birinchisi {\displaystyle n\to \infty } da nolga yaqinlashadi, {\displaystyle a\_{n}B\_{n}\to 0}.

Har bir *k* uchun, {\displaystyle |B\_{k}(a\_{k}-a\_{k+1})|\leq M|a\_{k}-a\_{k+1}|} munosabat o'rinli. Ammo, agar {\displaystyle \{a\_{n}\}} kamayuvchi bo'lsa,



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{\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}),}

{\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 {\displaystyle M(a\_{1}-a\_{n+1})} ga teng bo'ladi va shuning uchun {\displaystyle n\to \infty } da {\displaystyle Ma\_{1}} ga yaqinlashadi. Shunday qilib, {\textstyle \sum \_{k=1}^{\infty }M(a\_{k}-a\_{k+1})} yaqinlashuvchi bo'ladi. Shuningdek, agar {\displaystyle \{a\_{n}\}} o'suvchi bo'lsa,



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{\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}),}

{\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 {\displaystyle -M(a\_{1}-a\_{n+1})} ga teng bo'lib, {\displaystyle n\to \infty } da {\displaystyle -Ma\_{1}} ga yaqinlashadi. Shunday qilib, yana, {\textstyle \sum \_{k=1}^{\infty }M(a\_{k}-a\_{k+1})} yaqinlashuvchi bo'ladi.

Natijada, {\textstyle \sum \_{k=1}^{\infty }|B\_{k}(a\_{k}-a\_{k+1})|} to'g'ridan- to'g'ri *taqqoslash alomati* natijasida yaqinlashuvchi bo'ladi. {\textstyle \sum \_{k=1}^{\infty }B\_{k}(a\_{k}-a\_{k+1})} qator esa *absolyut yaqinlashish alomati* natijasida yaqinlashadi. Shuning uchun {\displaystyle S\_{n}} yaqinlashuvchi bo'ladi.

Ilovalar
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Dirixle alomatining xususiy holatlaridan biri bu amalda ko'proq qo'llaniladigan o'zgaruvchan qatorlarning yaqinlashish alomatidir



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{\displaystyle b\_{n}=(-1)^{n}\Longrightarrow \left|\sum \_{n=1}^{N}b\_{n}\right|\leq 1.}

{\displaystyle b_{n}=(-1)^{n}\Longrightarrow \left|\sum _{n=1}^{N}b_{n}\right|\leq 1.}Yana bir natija shuki, agar {\displaystyle \{a\_{n}\}} nolga yaqinlashadigan kamayuvchi ketma-ketlik bo'lsa, u holda {\textstyle \sum \_{n=1}^{\infty }a\_{n}\sin n} qator yaqinlashuvchi bo'ladi.

Xosmas integrallar
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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
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Manbalar
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* 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.

Havolalar
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* Isbot PlanetMath.org saytida