Qhov Sib Txawv Ntawm Cov lej lej thiab Geometric Series

2024 Tus sau: Alex Aldridge | [email protected]. Kawg hloov kho: 2023-12-17 13:40

Arithmetic vs Geometric Series

Lub ntsiab lus lej ntawm cov koob muaj feem cuam tshuam nrog cov kab ke. Ib ntu yog ib qho kev txiav txim ntawm cov lej thiab tuaj yeem yog ib qho finite lossis ib qho tsis kawg. Ib ntu ntawm cov lej nrog qhov sib txawv ntawm ob lub ntsiab lus ua qhov tsis tu ncua yog lub npe hu ua arithmetic progression. Ib ntu uas muaj qhov tsis tu ncua ntawm ob tus lej txuas ntxiv yog hu ua geometric kev nce qib. Cov kev nce qib no tuaj yeem ua tiav lossis tsis kawg, thiab yog tias tsis kawg, cov ntsiab lus suav tau, lwm yam suav tsis tau.

Feem ntau, cov lej ntawm cov ntsiab lus hauv kev nce qib tuaj yeem txhais tau tias yog series. Qhov sib npaug ntawm qhov kev nce qib ntawm tus lej lej yog lub npe hu ua arithmetic series. Ib yam li ntawd, cov lej ntawm qhov kev nce qib geometric hu ua geometric series.

Ntau txog lej lej

Nyob hauv cov lej lej, cov lus ua tiav muaj qhov sib txawv tas li.

S_n =a₁ + a₂ + a _{3+ a}4 _{+ ⋯} a_i; where a² =a₁ + d, a₃ =a₂ + d, thiab lwm yam.

qhov sib txawv d yog lub npe hu ua qhov sib txawv, thiab n^th term yog muab los ntawm a_n =a ₁+ (n-1)d; qhov twg a_{1 yog thawj lo lus.}

Tus cwj pwm ntawm koob hloov raws li qhov sib txawv d. Yog tias qhov sib txawv sib txawv yog qhov zoo, qhov kev nce qib yuav ua rau muaj txiaj ntsig zoo, thiab yog tias qhov sib txawv tsis zoo nws nyiam mus rau qhov tsis zoo infinity.

Cov lej ntawm cov koob tuaj yeem tau los ntawm cov qauv yooj yim hauv qab no, uas yog thawj zaug tsim los ntawm Indian astronomer thiab lej Aryabhata.

S_n =n/2 (a₁+ a_n)=n/2 [2a_{1 + (n-1)d]}

Tus lej S_{n tuaj yeem ua tiav lossis tsis kawg, raws li tus lej ntawm nqe lus.}

Ntau txog Geometric Series

A geometric series yog ib series nrog cov quotient ntawm cov zauv tas mus li. Nws yog ib qho tseem ceeb series pom nyob rau hauv txoj kev kawm ntawm koob, vim hais tias ntawm cov khoom nws muaj.

S_n =ar + ar² + ar³ +⋯+ ar ⁿ =∑_i=1 arⁱ

Raws li qhov piv r, tus cwj pwm ntawm cov koob tuaj yeem faib raws li hauv qab no. r={|r|≥1 series diverges; r≤1 series converges}. Tsis tas li ntawd, yog r<0 series oscillates, i.e. series muaj alternating values.

Cov lej ntawm cov lej geometric tuaj yeem suav nrog siv cov qauv hauv qab no. S_n =a(1-r) / (1-r); qhov twg a yog thawj lub sij hawm thiab r yog qhov piv. Yog hais tias tus piv r≤1, lub series converges. Rau ib qho infinite series, tus nqi ntawm convergence yog muab los ntawm S_{n=a / (1-r).}

Geometric series muaj ntau daim ntawv thov hauv kev tshawb fawb lub cev, engineering, thiab kev lag luam

Qhov txawv ntawm Arithmetic thiab Geometric Series yog dab tsi?

• Ib tug lej lej yog ib qho uas muaj qhov sib txawv tas li ntawm ob nqe lus uas nyob ib sab.

• Ib tug geometric series yog ib tug series nrog ib tug tas mus li quotient ntawm ob successive nqe lus.

• Txhua tus lej lej tsis kawg yog ib txwm sib txawv, tab sis nyob ntawm qhov sib piv, cov kab geometric tuaj yeem ua ke lossis sib txawv.

• Cov kab geometric tuaj yeem muaj oscillation hauv qhov muaj nuj nqis; uas yog, cov lej hloov lawv cov cim hloov pauv, tab sis cov lej lej tsis tuaj yeem muaj oscillations.