Teorema Abelya rezultat teoriyi stepenevih ryadiv nazvanij na chest norvezkogo matematika Nilsa Abelya Zmist 1 Tverdzhennya 2 Dovedennya 3 Prikladi 3 1 Priklad 1 3 2 Priklad 2 4 Div takozh 5 PosilannyaTverdzhennyared Nehaj f x n 0 a n x n displaystyle textstyle f x sum n geqslant 0 a n x n nbsp stepenevij ryad z kompleksnimi koeficiyentami i radiusom zbizhnosti R displaystyle R nbsp Yaksho ryad n 0 a n R n displaystyle textstyle sum n geqslant 0 a n R n nbsp ye zbizhnim todi lim x R f x n 0 a n R n displaystyle lim x to R f x sum n geqslant 0 a n R n nbsp Dovedennyared Zaminoyu zminnih u x R displaystyle u x R nbsp mozhna vvazhati R 1 displaystyle R 1 nbsp Takozh neobhidnim pidborom a 0 displaystyle a 0 nbsp mozhna pripustiti a n 0 displaystyle textstyle sum a n 0 nbsp Poznachimo S n displaystyle S n nbsp chastkovi sumi ryadu a n displaystyle textstyle sum a n nbsp Zgidno z pripushennyam lim n S n 0 displaystyle lim n to infty S n 0 nbsp i potribno dovesti sho lim x 1 f x 0 displaystyle lim x to 1 f x 0 nbsp Rozglyanemo x 0 1 displaystyle x in 0 1 nbsp Todi prijnyavshi S 1 0 displaystyle S 1 0 nbsp n 0 N S n S n 1 x n n 0 N S n x n x n 1 S N x N 1 displaystyle sum n 0 N S n S n 1 x n sum n 0 N S n x n x n 1 S N x N 1 nbsp Zvidsi oderzhuyetsya f x 1 x n 0 S n x n displaystyle textstyle f x 1 x sum n 0 infty S n x n nbsp Dlya dovilnogo e gt 0 displaystyle varepsilon gt 0 nbsp isnuye naturalne chislo N 0 displaystyle N 0 nbsp sho S n e displaystyle S n leq varepsilon nbsp dlya vsih n gt N 0 displaystyle n gt N 0 nbsp tomu f x 1 x n 0 N 0 S n x n 1 x e n N 0 1 x n 1 x n 0 N 0 S n x n e x N 0 1 displaystyle vert f x vert leqslant 1 x left vert sum n 0 N 0 S n x n right vert 1 x varepsilon sum n N 0 1 infty x n 1 x left vert sum n 0 N 0 S n x n right vert varepsilon x N 0 1 nbsp Prava chastina pryamuye do e displaystyle varepsilon nbsp koli x displaystyle x nbsp pryamuye do 1 zokrema vona ye menshoyu 2 e displaystyle 2 varepsilon nbsp pri pryamuvanni x displaystyle x nbsp do 1 Prikladired Priklad 1red Vizmemo f x n 1 1 n 1 x n n ln 1 x displaystyle textstyle f x sum n geqslant 1 frac 1 n 1 x n n ln 1 x nbsp Oskilki ryad n 1 1 n 1 n displaystyle textstyle sum n geqslant 1 frac 1 n 1 n nbsp zbigayetsya mayemo lim x 1 f x ln 2 n 1 1 n 1 n displaystyle lim x to 1 f x ln 2 sum n geqslant 1 frac 1 n 1 n nbsp Priklad 2red Vizmemo g x n 0 1 n x 2 n 1 2 n 1 arctan x displaystyle textstyle g x sum n geqslant 0 frac 1 n x 2n 1 2n 1 arctan x nbsp Ryad n 0 1 n 2 n 1 displaystyle textstyle sum n geqslant 0 frac 1 n 2n 1 nbsp zbigayetsya tomu lim x 1 g x arctan 1 p 4 n 0 1 n 2 n 1 displaystyle lim x to 1 g x arctan 1 frac pi 4 sum n geqslant 0 frac 1 n 2n 1 nbsp Div takozhred Spisok ob yektiv nazvanih na chest Nilsa Genrika AbelyaPosilannyared Abelya teoremi Vebsajt Velikoyi ukrayinskoyi enciklopediyi ukr Abel summability na PlanetMath angl a more general look at Abelian theorems of this type Weisstein Eric W Abel s Convergence Theorem angl na sajti Wolfram MathWorld Otrimano z https uk wikipedia org w index php title Teorema Abelya analiz amp oldid 42818985