Nazdikshavii mafhumi matematiki

Nazdikshavi, mafhumi matematikiest, ki hudud doshtani buzurgii tagyiryobandaro ifoda mekunad. Ba hamin mano dar borai Nazdikshavii paydarpai, Nazdikshavii kasri befosila, Nazdikshavii zarbi beokhir, Nazdikshavii integral va gayra sukhan rondan mumkin ast.

Nazdikshavii paydarpaii {a_n} (p = 1, 2, …) hududi okhirnok doshtani paydarpay, yane lim a_p = a; Nazdikshavii qatori _k

hududi okhirnok doshtani paydarpaii summahoi khususii qator (nig.

Qator) yane lim S_n=s (S_n = _k  , n =  1,  2, …); Nazdikshavii zarbi beokhir b₁ ∙ b₂∙… b_n  hududi okhirnoki gayrisifri doshtani paydarpaii zarbhoi okhirnok R_p = b₁ ∙ b₂ ∙… ∙ _p, p = 1, 2, …; Nazdikshavii integrali {x)dx az funkciyai f(kh), ki dar oar guna porchai okhirnoki [a, b] integronidashavanda meboshad, hududi okhirnok doshtani integralhoro hangomi b→+ , yane integrali gayrikhosi +

{x)dx – ro ifoda mekunad.

Khosiyati Nazdikshavii in yo on obekthoi matematiki ham dar masalahoi nazariyavi va ham dar masalahoi amalii matematika roli muhim mebozad. Masalan, aksar buzurgi yo funksiyahoro bo vositai qatorhoi nazdiksha­vanda ifoda mekunand, asosi logarifmi naturali e-ro ba qatori nazdikshavanda in tavr pahn mekunand:

e= 1+  +  +  + ….+ + …,

funksiyai sin kh boshad, baroi haman kh bo qatori nazdikshavandai

sin x= x-  +  –  + … +) ∙  + …

 

ifoda meshavad. Hamin tavr, qatorhoro baroi taqribi hisob kardani buzurgi va funksiyahoi gunogun istifoda burdan mumkin budaast. Ba­roi in summai yakchand uzvhoi avva­li qatorro giriftan kifoya ast. Miqdori uzvho har qadar bisyor boshand, buzurgii matlub niz hamon qadar sahehtar meshavad. Baroi ha- mon yak buzurgi va yo funksiya qatorhoi gunogune mavjudand, ki summai onho ba in buzurgi va yo funksiya bapobap ast, masalan,

ln  =  –  ∙   +  ∙  –  ∙  + . . . +    + . . . ,

 

ln  =  (1+  ∙  +  ∙  + . . . +  ∙  + . . . ).

 

Hangomi hisobu kitobi amali bo maqsadi sarfa namudani miqdori amalho (dar barobari in sarfa na­mudani vaqt va kam kardani khatoho) qatorero intikhob namudan lozim ast, ki on “nisbatan teztar nazdikshavanda” boshad.

 

Bigzor du qatori nazdikshavandai

 

doda shuda boshand, bo   va   muvofiqan baqiyahoi qatori yakum va duyumro ishora mekunem. Agar

 

 

boshad, qatori yakum nisbat ba qatori duyum teztar nazdikshavanda nomida meshavad.

Masalan, qatori

1+

nisbat ba qatori

 

teztar napdikshavanda meboshad.

Mafhumi Nazdikshavi dar halli bisyor muodilaho (algebravi, differensiali, integrali), az jumla hangomi justujui khalhoi adadii taqribii muodilaho ahamiyati kalon dorad. Masalan, bo yorii usuli pay dar pay nazdikkuni paydarpaii funksiyahoero. ki ba halli muvofiqi muodilai differensiali nazdikshavandaand, hosil kardan mumkin ast.

Dar tahlili matematiki namudhoi gunoguni Nazdikshavi paydarpaihoi funk­siyahoi { (x)} ba funkciyai f{x) dar yagon majmui M muoina karda me­shavad. Agar baroi har yak nuqtai az M  (kh₀) = f (kh₀) boshad, dar borai Nazdikshavi dar har yak nuqta sukhan meronand (agar in barobari tanho baroi nuqtahoi maqmui cheni sinfiro tashkilkunanda joiz naboshad, dar borai Nazdikshavi qarib dar hama qo sukhan meronand). Nazdikshavi yakchand khu- susiyathoi nokulay ham dorad (masalan, paydarpaii funksiyahoi befosila dar har yak nuqta ba funksiyai kanishdor nazdik shudanash mumkin ast, az Nazdikshavvi funksiyahoi (kh) ba f(x) dar har yak nuqta dar holati umumi Nazdikshavii integralho az  (kh) ba integral az f(x)  barnameoyad va gayra). Vobasta ba in mafhumi Nazdikshavii muntazam jori karda shudaast. Ba monan- di hamin, dar nazariyai muodilahoi integrali mafhumi Nazdikshavii miyonai kvadrati jori gardidaast. Mafhumi Nazdikshaviro dar nazariyai ehtimoliyat, tahlili funksionali niz istifoda mebarand.

Ad.: I l  i i V. A., P o 8 ya ya k E. G., Osnovi matematicheskogo analiza. 3 izd., t. 1—2, M., 1107.1—73; Kudryavtsev L. D.. Matematicheskiy analiz, 2 izd., t. 1—2, M., 1970; Nikolskiy S. M., Kurs matematicheskogo analiza, t. 1—2, M., 1973.