chislovie ryadi

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chislovie ryadi osnovnie ponyatiya chislovix ryadov. neobxodimie usloviya sxodimosti ryadov. teoremi sravneniya polojitelnix ryadov. dostatochnie usloviya sxodimosti ryadov: znak dalambera, radikalniy i tselochislenniy znaki koshi. znakoperemennie ryadi. teorema leybnitsa. absolyutniy i uslovno sxodyashiysya ryad. linii teylora i maklorena plan: 1. osnovnie ponyatiya i neobxodimie usloviya sxodimosti 2. dostatochnie priznaki sxodimosti dlya polojitelnix ryadov 3. znakoperemennie ryadi i stepennie razlojeniya 1. osnovnie ponyatiya i neobxodimie usloviya sxodimosti chislovie ryadi yavlyayutsya odnim iz fundamentalnix ponyatiy matematicheskogo analiza. ryad predstavlyaet soboy beskonechnuyu summu chisel, kotoraya zapisivaetsya kak: n=1∑∞an=a1+a2+a3+⋯+an+… gde an — eto obshiy chlen ryada (ili n-y chlen ryada). osnovnie ponyatiya obshiy chlen ryada (an) formula obshego chlena ryada yavlyaetsya opredelyayushey xarakteristikoy. znanie an pozvolyaet vichislit lyuboy chlen ryada i, chto samoe glavnoe, issledovat ego sxodimost. chastichnaya summa ryada (sn) chastichnaya summa sn — eto summa pervix n chlenov ryada: sn=a1+a2+⋯+an=n=1∑nan posledovatelnost chastichnix summ {sn} yavlyaetsya klyuchevim elementom dlya opredeleniya sxodimosti ryada. sxodimost …

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utsya putem sopostavleniya issleduemogo ryada s drugim ryadom, sxodimost kotorogo uje izvestna. 1. priznak sravneniya v obichnoy forme (neravenstvo) teorema: pust dani dva polojitelnix ryada ∑an i ∑bn. 1. esli an≤bn nachinaya s nekotorogo nomera n0, i ryad ∑bn sxoditsya, to i ryad ∑an sxoditsya. 2. esli an≥bn nachinaya s nekotorogo nomera n0, i ryad ∑bn rasxoditsya, to i ryad ∑an rasxoditsya. v kachestve etalonnix (sravnivaemix) ryadov chasto ispolzuyut: · ryad dirixle (obobshenniy garmonicheskiy ryad): ∑n=1∞np1. sxoditsya, esli p>1; rasxoditsya, esli p≤1. · geometricheskiy ryad: ∑n=0∞qn. sxoditsya, esli ∣q∣ 1 (ili l=∞), ryad rasxoditsya. 3. esli l=1, priznak ne daet otveta (trebuetsya ispolzovat drugoy priznak). priznak d’alambera osobenno effektiven dlya ryadov, soderjashix faktoriali (n!) i stepennie funktsii (an), tak kak pri vzyatii otnosheniya proisxodit silnoe sokrashenie. 2. radikalniy priznak koshi (cauchy's root test) etot priznak ispolzuet predel kornya n-y stepeni iz obshego chlena. teorema: pust dlya polojitelnogo ryada ∑an sushestvuet …

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∑np1. vivod: pri issledovanii polojitelnix ryadov rekomenduetsya sleduyushiy poryadok deystviy: 1) proverka neobxodimogo usloviya. 2) ispolzovanie priznakov d’alambera ili koshi (dlya ryadov s faktorialami/stepenyami). 3) ispolzovanie priznakov sravneniya (dlya sravneniya s ryadami dirixle). 4) ispolzovanie integralnogo priznaka (dlya funktsiy, legko integriruemix). 3. znakoperemennie ryadi i stepennie razlojeniya znakoperemennie ryadi – eto ryadi, chleni kotorix mogut prinimat kak polojitelnie, tak i otritsatelnie znacheniya. oni delyatsya na znakochereduyushiesya i ryadi s proizvolnim cheredovaniem znakov. znakochereduyushiesya ryadi znakochereduyushiysya ryad – eto chastniy sluchay znakoperemennogo ryada, gde znaki sosednix chlenov chereduyutsya strogo po ocheredi: n=1∑∞(−1)n−1an=a1−a2+a3−a4+… gde an>0. teorema leybnitsa (leibniz's test) eta teorema daet prostoe dostatochnoe uslovie sxodimosti dlya znakochereduyushixsya ryadov. teorema (priznak leybnitsa): znakochereduyushiysya ryad ∑n=1∞(−1)n−1an sxoditsya, esli vipolnyayutsya dva usloviya: 1. chleni ryada monotonno ubivayut po absolyutnoy velichine: a1≥a2≥a3≥⋯≥an≥… 2. obshiy chlen ryada stremitsya k nulyu: limn→∞an=0. otsenka ostatka ryada leybnitsa: esli ryad sxoditsya po priznaku leybnitsa, to ego ostatok rn=s−sn (raznost …

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ki absolyutnoy sxodimosti ispolzuyutsya priznaki d’alambera i koshi, primenennie k ryadu ∑∣an∣. 2. uslovnaya sxodimost ryad ∑an nazivaetsya uslovno sxodyashimsya, esli sam ryad sxoditsya, a ryad, sostavlenniy iz absolyutnix velichin, rasxoditsya (t.e. ∑an sxoditsya, a ∑∣an∣ rasxoditsya). uslovno sxodyashiesya ryadi obladayut paradoksalnim svoystvom: ix summa mojet izmenitsya (i daje stat lyuboy napered zadannoy) pri izmenenii poryadka slagaemix (teorema rimana). ryadi teylora i maklorena ryadi teylora i maklorena yavlyayutsya klyuchevim zvenom, soedinyayushim chislovie ryadi s funktsionalnimi (stepennimi) ryadami. stepennoy ryad stepennoy ryad — eto funktsionalniy ryad spetsialnogo vida: n=0∑∞cn(x−x0)n=c0+c1(x−x0)+c2(x−x0)2+… gde x0 — tsentr ryada, a cn — koeffitsienti. oblast sxodimosti stepennogo ryada vsegda yavlyaetsya intervalom sxodimosti, simmetrichnim otnositelno x0. ryad teylora esli funktsiya f(x) beskonechno differentsiruema v okrestnosti tochki x0, to ee ryad teylora opredelyaetsya kak: f(x)≈n=0∑∞n!f(n)(x0)(x−x0)n koeffitsienti ryada cn=n!f(n)(x0) odnoznachno opredelyayutsya proizvodnimi funktsii v tsentre razlojeniya. ryad maklorena ryad maklorena — eto chastniy sluchay ryada teylora pri x0=0: f(x)≈n=0∑∞n!f(n)(0)xn ryadi …

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skonechnix ryadov, otkrivaya moshnie metodi approksimatsii.

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chislovie ryadi osnovnie ponyatiya chislovix ryadov. neobxodimie usloviya sxodimosti ryadov. teoremi sravneniya polojitelnix ryadov. dostatochnie usloviya sxodimosti ryadov: znak dalambera, radikalniy i tselochislenniy znaki koshi. znakoperemennie ryadi. teorema leybnitsa. absolyutniy i uslovno sxodyashiysya ryad. linii teylora i maklorena plan: 1. osnovnie ponyatiya i neobxodimie usloviya sxodimosti 2. dostatochnie priznaki sxodimosti dlya polojitelnix ryadov 3. znakoperemennie ryadi i stepennie razlojeniya 1. osnovnie ponyatiya i neobxodimie usloviya sxodimosti chislovie ryadi yavlyayutsya odnim iz fundamentalnix ponyatiy matematicheskogo analiza. ryad predstavlyaet soboy beskonechnuyu summu chisel, kotoraya zapisivaetsya kak: n=1∑∞an=a1+a2+a3+⋯+an+… gde an — eto obshiy chlen ryad...

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