Serednye arifmetiko geometrichne ce spilna granicya dvoh poslidovnostej serednoyi arifmetichnoyi ta serednoyi geometrichnoyi dvoh zadanih chisel a ta b Dovedennya RedaguvatiNehaj nam zadano dva dodatnih chisla a ta b prichomu a gt b displaystyle a gt b nbsp Utvorimo yih serednye arifmetichne ta serednye geometrichne a 1 a b 2 b 1 a b displaystyle a 1 tfrac a b 2 qquad b 1 sqrt ab nbsp Vidomo sho pershe serednye bilshe za druge a b 2 a b 1 2 a 2 a b b a b 2 2 gt 0 displaystyle tfrac a b 2 sqrt ab tfrac 1 2 a 2 sqrt ab b tfrac sqrt a sqrt b 2 2 gt 0 nbsp V toj zhe chas voni mistyatsya mizh zadanimi chislami a gt a 1 gt b 1 gt b displaystyle a gt a 1 gt b 1 gt b nbsp Yaksho chisla a n displaystyle a n nbsp ta b n displaystyle b n nbsp vzhe viznacheni to a n 1 displaystyle a n 1 nbsp ta b n 1 displaystyle b n 1 nbsp viznachayutsya za formulami a n 1 a n b n 2 b n 1 a n b n displaystyle a n 1 tfrac a n b n 2 qquad b n 1 sqrt a n b n nbsp i yak i vishe a n gt a n 1 gt b n 1 gt b n displaystyle a n gt a n 1 gt b n 1 gt b n nbsp Takim chinom skladayutsya dvi varianti a n displaystyle a n nbsp ta b n displaystyle b n nbsp persha z yakih ye spadnoyu a insha zrostayuchoyu na zustrich odna odnij V toj zhe chas a gt a n gt b n gt b displaystyle a gt a n gt b n gt b nbsp Tak sho obidvi varianti obmezheni i vidpovidno obidvi pryamuyut do kincevih granic a lim a n b lim b n displaystyle alpha lim a n qquad beta lim b n nbsp Yaksho v rivnyannia n 1 a n b n 2 displaystyle a n 1 tfrac a n b n 2 nbsp perejti do granic to otrimayemoa a b 2 displaystyle alpha tfrac alpha beta 2 nbsp zvidkilya a b displaystyle alpha beta nbsp Takim chinom obidvi poslidovnosti pryamuyut do spilnoyi granici m a b displaystyle mu a b nbsp Literatura RedaguvatiGrigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1300 s ukr nbsp Ce nezavershena stattya z matematiki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Serednye arifmetiko geometrichne amp oldid 40456086
