Bulo zaproponovano priyednati cyu stattyu abo rozdil do Ryad matematika ale mozhlivo ce varto dodatkovo obgovoriti Propoziciya z kvitnya 2015 Bulo zaproponovano priyednati cyu stattyu abo rozdil do Oznaki zbizhnosti ale mozhlivo ce varto dodatkovo obgovoriti Propoziciya z kvitnya 2015 Chislovij ryad ryad elementami yakogo ye chisla Nehaj a n n N displaystyle a n n in mathbb N deyaka chislova poslidovnist Dlya kozhnogo n N displaystyle n in mathbb N viznachena skinchenna suma S n a 1 a 2 a n displaystyle S n a 1 a 2 cdots a n Dvi chislovi poslidovnosti a n n N displaystyle a n n in mathbb N ta S n n N displaystyle S n n in mathbb N nazivayutsya chislovim ryadom i poznachayutsya a 1 a 2 a n n 1 a n displaystyle a 1 a 2 cdots a n cdots sum n 1 infty a n Chislo a n displaystyle a n nazivayetsya n tim chlenom a chislo S n displaystyle S n n toyu chastkovoyu sumoyu ryadu Yaksho poslidovnist chastkovih sum S n displaystyle S n zbigayetsya do deyakogo chisla S displaystyle S div Granicya chislovoyi poslidovnosti to chislovij ryad nazivayetsya zbizhnim a chislo S displaystyle S nazivayetsya sumoyu cogo ryadu i poznachayetsya S n 1 a n displaystyle S sum n 1 infty a n Yaksho zh skinchennoyi granici ne isnuye to chislovij ryad nazivayetsya rozbizhnim Zmist 1 Teoremi 2 Vlastivosti zbizhnih ryadiv 3 Divis takozh 4 Literatura 5 PosilannyaTeoremi RedaguvatiTeorema 01 Yaksho chislovij ryad n 1 a n displaystyle sum n 1 infty a n zbigayetsya toa n 0 displaystyle a n rightarrow 0 n displaystyle n rightarrow infty Dovedennya displaystyle vartriangleright Dijsno oskilki a n S n S n 1 displaystyle a n S n S n 1 n 2 displaystyle n geqslant 2 ta S n S R displaystyle S n rightarrow S in mathbb R n displaystyle n rightarrow infty to a n S S 0 displaystyle a n rightarrow S S 0 n displaystyle n rightarrow infty displaystyle vartriangleleft Teorema 02 Yaksho chislovij ryad n 1 a n displaystyle sum n 1 infty a n zbigayetsya toa n 1 a n 2 a 2 n 0 displaystyle a n 1 a n 2 cdots a 2n rightarrow 0 n displaystyle n rightarrow infty Dovedennya displaystyle vartriangleright Rozglyanemo a n 1 a n 2 a 2 n S 2 n S n S S 0 displaystyle a n 1 a n 2 cdots a 2n S 2n S n rightarrow S S 0 n displaystyle n rightarrow infty displaystyle vartriangleleft Teoremi 01 ta 02 dayut neobhidni umovi zbizhnosti ryadu 1 Priklad 01 Ryadi1 1 1 1 displaystyle 1 1 1 cdots 1 cdots 2 1 1 1 1 n 1 displaystyle 1 1 1 cdots 1 n 1 cdots 3 ye rozbizhnimi zgidno z teoremoyu 01 Dijsno a n 1 0 displaystyle a n 1 nrightarrow 0 n displaystyle n rightarrow infty u vipadku ryadu 1 ta a n 1 n 1 0 displaystyle a n 1 n 1 nrightarrow 0 u vipadku ryadu 2 Priklad 02 Geometrichnij ryad dlya x R displaystyle x in mathbb R maye viglyad1 x x 2 x n displaystyle 1 x x 2 cdots x n cdots 4 Jogo chastkova sumaS n n x 1 1 x n 1 x x 1 displaystyle S n begin cases n amp x 1 frac 1 x n 1 x amp x neq 1 end cases dlya n 1 displaystyle n geqslant 1 displaystyle vartriangleright Yaksho x lt 1 displaystyle x lt 1 to x n 0 displaystyle x n rightarrow 0 n displaystyle n rightarrow infty Tobto pri x lt 1 displaystyle x lt 1 ryad 4 zbigayetsya do sumi 1 1 x displaystyle frac 1 1 x 1 x x 2 x n 1 1 x displaystyle 1 x x 2 cdots x n cdots frac 1 1 x x lt 1 displaystyle x lt 1 Pri x 1 displaystyle x geqslant 1 poslidovnist S n n 1 displaystyle S n colon n geqslant 1 skinchennoyi granici ne maye otzhe pri x 1 displaystyle x geqslant 1 ryad 4 rozbigayetsya displaystyle vartriangleleft Priklad 03 Dovedemo sho1 1 2 1 2 3 1 3 4 1 n n 1 1 displaystyle frac 1 1 cdot 2 frac 1 2 cdot 3 frac 1 3 cdot 4 cdots frac 1 n n 1 cdots 1 displaystyle vartriangleright Dijsno dlya n 1 displaystyle n geqslant 1 S n 1 1 2 1 2 3 1 3 4 1 n n 1 1 1 2 1 2 1 3 1 3 1 4 1 n 1 n 1 1 1 n 1 displaystyle S n frac 1 1 cdot 2 frac 1 2 cdot 3 frac 1 3 cdot 4 cdots frac 1 n n 1 1 frac 1 2 frac 1 2 frac 1 3 frac 1 3 frac 1 4 cdots frac 1 n frac 1 n 1 1 frac 1 n 1 Otzhe S n 1 displaystyle S n rightarrow 1 n displaystyle n rightarrow infty displaystyle vartriangleleft Priklad 04 Garmonichnij ryad maye viglyad1 1 2 1 3 1 n displaystyle 1 frac 1 2 frac 1 3 cdots frac 1 n cdots displaystyle vartriangleright Dovedemo sho cej ryad rozbigayetsya Vikoristovuyuchi teoremu 02 pri n 1 displaystyle n geqslant 1 matimemoS 2 n S n 1 n 1 1 n 2 1 2 n n 1 2 n 1 2 displaystyle S 2n S n frac 1 n 1 frac 1 n 2 cdots frac 1 2n geqslant n frac 1 2n frac 1 2 Takim chinom S 2 n S n 0 displaystyle S 2n S n nrightarrow 0 n displaystyle n rightarrow infty Oskilki poslidovnist S n n 1 displaystyle S n colon n geqslant 1 zrostaye ta ne maye granici to S n displaystyle S n rightarrow infty n displaystyle n rightarrow infty Prote zrostannya S displaystyle S iz zrostannyam n displaystyle n vidbuvayetsya duzhe povilno L Ejler pidrahuvav sho S 1000000 14 displaystyle S 1000000 approx 14 Varto takozh zvernuti uvagu sho chleni garmonijnogo ryadu pryamuyut do nulya pri n displaystyle n rightarrow infty tobto neobhidna umova zbizhnosti vikonuyetsya displaystyle vartriangleleft Vlastivosti zbizhnih ryadiv Redaguvati1 Nehaj ryad n 1 a n displaystyle sum n 1 infty a n zbigayetsya do sumi S displaystyle S Todi dlya bud yakogo c R displaystyle c in mathbb R ryad n 1 c a n displaystyle sum n 1 infty ca n tezh zbigayetsya i maye sumu c S displaystyle cS tobto n 1 c a n c n 1 a n displaystyle sum n 1 infty ca n c sum n 1 infty a n displaystyle vartriangleright Dovedennya viplivaye z oznachen displaystyle vartriangleleft 2 Nehaj ryadi n 1 a n displaystyle sum n 1 infty a n ta n 1 a n displaystyle sum n 1 infty a n zbigayutsya do sum S displaystyle S ta S displaystyle S vidpovidno Todi ryad n 1 a n a n displaystyle sum n 1 infty a n a n zbigayetsya do sumi S S displaystyle S S tobto n 1 a n a n n 1 a n n 1 a n displaystyle sum n 1 infty a n a n sum n 1 infty a n sum n 1 infty a n Oznachennya Dlya ryadua 1 a 2 a n displaystyle a 1 a 2 cdots a n cdots 1 ta chisla m N displaystyle m in mathbb N ryada m 1 a m 2 a n displaystyle a m 1 a m 2 cdots a n cdots 2 nazivayetsya zalishkom vihidnogo ryadu Yaksho ryad 2 zbigayetsya to r m displaystyle r m suma zalishku 3 Yaksho ryad 1 zbigayetsya do sumi S displaystyle S to zbigayetsya bud yakij jogo zalishok prichomu m N S S m r m displaystyle forall m in mathbb N colon S S m r m Yaksho dlya deyakogo n N displaystyle n in mathbb N zbigayetsya zalishok 2 to ryad 1 zbigayetsya 4 Kriterij Koshi zbizhnosti chislovogo ryadu Dlya togo shob ryad 1 zbigavsya neobhidno i dostatno shob e gt 0 N N n N p N displaystyle forall varepsilon gt 0 exists N in mathbb N forall n geqslant N forall p in mathbb N colon a n 1 a n 2 a n p lt e displaystyle a n 1 a n 2 cdots a n p lt varepsilon displaystyle vartriangleright Cej kriterij yavlyaye soboyu kriterij Koshi dlya chislovoj poslidovnosti S n n 1 displaystyle S n colon n geqslant 1 displaystyle vartriangleleft Divis takozh RedaguvatiOznaki zbizhnosti Ryad matematika Literatura RedaguvatiDorogovcev A Ya Matematicheskij analiz Spravochnoe posobie K Visha shk Golovnoe izd vo 1985 Matematicheskaya enciklopediya V pyati tomah Tom 4 Sovetskaya enciklopediya 1984 S T Zavalo 1972 Elementi analizu Algebra mnogochleniv Kiyiv Radyanska shkola Posilannya RedaguvatiRyadi Visha matematika v prikladah i zadachah Klepko V Yu Golec V L 2 ge vidannya K Centr uchbovoyi literaturi 2009 S 496 594 s Otrimano z https uk wikipedia org w index php title Chislovij ryad amp oldid 34493219
