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 nbsp zbigayetsya toa n 0 displaystyle a n rightarrow 0 nbsp n displaystyle n rightarrow infty nbsp Dovedennya displaystyle vartriangleright nbsp Dijsno oskilki a n S n S n 1 displaystyle a n S n S n 1 nbsp n 2 displaystyle n geqslant 2 nbsp ta S n S R displaystyle S n rightarrow S in mathbb R nbsp n displaystyle n rightarrow infty nbsp to a n S S 0 displaystyle a n rightarrow S S 0 nbsp n displaystyle n rightarrow infty nbsp displaystyle vartriangleleft nbsp Teorema 02 Yaksho chislovij ryad n 1 a n displaystyle sum n 1 infty a n nbsp zbigayetsya toa n 1 a n 2 a 2 n 0 displaystyle a n 1 a n 2 cdots a 2n rightarrow 0 nbsp n displaystyle n rightarrow infty nbsp Dovedennya displaystyle vartriangleright nbsp 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 nbsp n displaystyle n rightarrow infty nbsp displaystyle vartriangleleft nbsp Teoremi 01 ta 02 dayut neobhidni umovi zbizhnosti ryadu 1 Priklad 01 Ryadi1 1 1 1 displaystyle 1 1 1 cdots 1 cdots nbsp 2 1 1 1 1 n 1 displaystyle 1 1 1 cdots 1 n 1 cdots nbsp 3 ye rozbizhnimi zgidno z teoremoyu 01 Dijsno a n 1 0 displaystyle a n 1 nrightarrow 0 nbsp n displaystyle n rightarrow infty nbsp u vipadku ryadu 1 ta a n 1 n 1 0 displaystyle a n 1 n 1 nrightarrow 0 nbsp u vipadku ryadu 2 Priklad 02 Geometrichnij ryad dlya x R displaystyle x in mathbb R nbsp maye viglyad1 x x 2 x n displaystyle 1 x x 2 cdots x n cdots nbsp 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 nbsp dlya n 1 displaystyle n geqslant 1 nbsp displaystyle vartriangleright nbsp Yaksho x lt 1 displaystyle x lt 1 nbsp to x n 0 displaystyle x n rightarrow 0 nbsp n displaystyle n rightarrow infty nbsp Tobto pri x lt 1 displaystyle x lt 1 nbsp ryad 4 zbigayetsya do sumi 1 1 x displaystyle frac 1 1 x nbsp 1 x x 2 x n 1 1 x displaystyle 1 x x 2 cdots x n cdots frac 1 1 x nbsp x lt 1 displaystyle x lt 1 nbsp Pri x 1 displaystyle x geqslant 1 nbsp poslidovnist S n n 1 displaystyle S n colon n geqslant 1 nbsp skinchennoyi granici ne maye otzhe pri x 1 displaystyle x geqslant 1 nbsp ryad 4 rozbigayetsya displaystyle vartriangleleft nbsp 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 nbsp displaystyle vartriangleright nbsp Dijsno dlya n 1 displaystyle n geqslant 1 nbsp 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 nbsp Otzhe S n 1 displaystyle S n rightarrow 1 nbsp n displaystyle n rightarrow infty nbsp displaystyle vartriangleleft nbsp 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 nbsp displaystyle vartriangleright nbsp Dovedemo sho cej ryad rozbigayetsya Vikoristovuyuchi teoremu 02 pri n 1 displaystyle n geqslant 1 nbsp 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 nbsp Takim chinom S 2 n S n 0 displaystyle S 2n S n nrightarrow 0 nbsp n displaystyle n rightarrow infty nbsp Oskilki poslidovnist S n n 1 displaystyle S n colon n geqslant 1 nbsp zrostaye ta ne maye granici to S n displaystyle S n rightarrow infty nbsp n displaystyle n rightarrow infty nbsp Prote zrostannya S displaystyle S nbsp iz zrostannyam n displaystyle n nbsp vidbuvayetsya duzhe povilno L Ejler pidrahuvav sho S 1000000 14 displaystyle S 1000000 approx 14 nbsp Varto takozh zvernuti uvagu sho chleni garmonijnogo ryadu pryamuyut do nulya pri n displaystyle n rightarrow infty nbsp tobto neobhidna umova zbizhnosti vikonuyetsya displaystyle vartriangleleft nbsp Vlastivosti zbizhnih ryadiv Redaguvati1 Nehaj ryad n 1 a n displaystyle sum n 1 infty a n nbsp zbigayetsya do sumi S displaystyle S nbsp Todi dlya bud yakogo c R displaystyle c in mathbb R nbsp ryad n 1 c a n displaystyle sum n 1 infty ca n nbsp tezh zbigayetsya i maye sumu c S displaystyle cS nbsp 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 nbsp displaystyle vartriangleright nbsp Dovedennya viplivaye z oznachen displaystyle vartriangleleft nbsp 2 Nehaj ryadi n 1 a n displaystyle sum n 1 infty a n nbsp ta n 1 a n displaystyle sum n 1 infty a n nbsp zbigayutsya do sum S displaystyle S nbsp ta S displaystyle S nbsp vidpovidno Todi ryad n 1 a n a n displaystyle sum n 1 infty a n a n nbsp zbigayetsya do sumi S S displaystyle S S nbsp 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 nbsp Oznachennya Dlya ryadua 1 a 2 a n displaystyle a 1 a 2 cdots a n cdots nbsp 1 ta chisla m N displaystyle m in mathbb N nbsp ryada m 1 a m 2 a n displaystyle a m 1 a m 2 cdots a n cdots nbsp 2 nazivayetsya zalishkom vihidnogo ryadu Yaksho ryad 2 zbigayetsya to r m displaystyle r m nbsp suma zalishku 3 Yaksho ryad 1 zbigayetsya do sumi S displaystyle S nbsp 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 nbsp Yaksho dlya deyakogo n N displaystyle n in mathbb N nbsp 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 nbsp a n 1 a n 2 a n p lt e displaystyle a n 1 a n 2 cdots a n p lt varepsilon nbsp displaystyle vartriangleright nbsp Cej kriterij yavlyaye soboyu kriterij Koshi dlya chislovoj poslidovnosti S n n 1 displaystyle S n colon n geqslant 1 nbsp displaystyle vartriangleleft nbsp Divis takozh RedaguvatiOznaki zbizhnosti Ryad matematika Literatura RedaguvatiGrigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1300 s ukr Dorogovcev 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 40109548
