Kupitilira zojambula ndi katundu

kupitilira zojambula ndi zofunika masamu ndi sayansi, ndipo ntchito tanthauzo, chifukwa ali ndi kuchuluka waukulu kwambiri, mpaka mu masamu apamwamba, mwachitsanzo, mu chiphunzitso ija. Mfundo yoyamba pa patsogolo anabwera kwa ife ku Egypt, makamaka mu mawonekedwe a vuto odziwika wa Rhind amtunduwu anthu asanu ndi amphaka seveni. Kusiyana kwa ntchito anabwerezedwanso nthawi zambiri panthawi zosiyanasiyana mitundu ina. Ngakhale Velikiy Leonardo Pizansky, lotchedwa Fibonacci (XIII c.), Ndinayankhula kwa iye mu "Bukhu la Abacus."

Kotero kuti patsogolo zojambula ali ndi mbiri yakale. Bwaloli zinayendera n'zosangalatsa kumva ndi nonzero woyamba, ndipo aliyense wotsatira, kuyambira ndi wachiwiri anatsimikiza ndi kuchulukitsa m'mbuyomo zisadzachitikenso chilinganizo nthaŵi zonse nonzero chiwerengero wotchedwa chimene chachititsa kupitirira (izo kawirikawiri anasankha ntchito kalata Q).
Mwachidziwikire, angapezeke ndi kugawa lililonse akuti wotsatira wa zinayendera kuti m'mbuyomo, i.e. z 2: z 1 = ... = zn: z N-1 = .... Choncho, ntchito patsogolo kwambiri (zn) wokwanira kuti amadziwa kufunika kwa Mauwa wa chimene chachititsa ndi Y 1 Q.

Mwachitsanzo, tiyeni z 1 = 7, Q = - 4 (Q <0), ndiye zotsatirazi zojambula kupitilira analandira 7 - 28, 112 - 448, .... Monga mukuonera, akuyambira chifukwa si popanda kusintha mphamvu.

Kumbukirani kuti ndondomeko owonjezera a wonyong'onya (kuwonjezeka / kucheperachepera) pamene wina m'banjamo kutsatira kwambiri / zosakwana imodzi yapita. Mwachitsanzo, akuyambira 2, 5, 9, ..., ndipo -10, -100, -1000, ... - popanda kusintha mphamvu, kachiwiri - ndi kucheperachepera zojambula kupitirira.

Ngati kumene Q = 1, ziwalo zonse amapezeka kuti, ndipo akutchedwa kupitilira zonse.

Akuyambira anali kudwala wa mtundu uwu, ayenera kukwaniritsa izi chikhalidwe zofunika wokwanira, akuti: kuyambira wachiwiri, yense wa mamembala ake ayenera kukhala zikutanthauza zojambula za anthu oyandikana nawo.

katundu Zimenezi zimathandiza pansi zina zoyandikana kupeza umasinthasintha akuti kupitirira.

N-TH akuti exponentially mosavuta anapeza mfundo: zn = z 1 * Q ^ (n-1), z osadziwa membala 1 ndi chimene chachititsa Q.

Popeza ndondomeko chiwerengero ali ndi ndalama, kenako kuwerengetsera wamba ochepa kutipatsa chilinganizo ndi kuwerengera ndalama za kupitilira woyamba wa anthu, akuti:

S N = - (zn * Q - z 1) / (1 - Q).

M'malo, mu njira yake phindu akuti zn z 1 * Q ^ (n-1) kupeza chachiwiri ndalama chilinganizo cha kupitilira: S N = - z1 * (Q ^ N - 1) / (1 - Q).

Lotereli zotsatirazi chidwi mfundo izi: phale opezeka zofukulidwa wa Babulo, kutanthauza VI. BC, lili mwapadera Uwerenge 1 + 2 + ... + 22 + 29 wofanana 2 mphamvu chakhumi opanda 1. kufotokozera za zodabwitsazi silinakwaniritsidwe wapezeka.

Tingaonenso wina wa katundu wa kupitilira zojambula - ntchito zonse anthu ake, spaced pa mtunda wofanana kuchokera kumalekezero a ndondomeko ya.

Chofunika kwambiri ndi mfundo za sayansi view, chinthu monga wopandamalire zojambula kupitirira ndi kuwerengetsa kuchuluka kwake. Kungoganiza kuti (yn) - ndi kupitirira zojambula ndi chimene chachititsa Q, kukhutiritsa chikhalidwe | Q | <1, kuchuluka zake anatchula malire kwa zomwe ife tikudziwa kale ndalama za anthu ake choyamba, ndi chosefukira kuchuluka kwa N, ndiye pa izo akuyandikira osawerengeka.

Pezani ndalamazo chifukwa cha ntchito njira:

S N = Y 1 / (1- Q).

Ndipo, monga zochitika zasonyeza, chifukwa kuphweka mwina kupitirira izi wabisika chachikulu ntchito kuthekera. Mwachitsanzo, ngati ife kumanga ndondomeko ya mabwalo monga mwa aligorivimu zotsatirazi, kulumikiza midpoints wa munthu yapita, ndiye iwo kupanga lalikulu wopandamalire zojambula kupitilira kukhala chimene chachititsa 1/2. Yemweyo kupitilira mawonekedwe ndi dera Triangles, analandira pa chilichonse siteji ya zomangamanga, ndipo ndalama zake ndi wofanana ndi dera lalikulu pachiyambi.