Qhov txawv ntawm Lub Hwj Chim Series thiab Taylor Series

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

Power Series vs Taylor Series

Hauv kev ua lej, qhov sib lawv liag tiag yog ib daim ntawv teev cov lej tiag. Raws li txoj cai, nws yog ib qho kev ua haujlwm los ntawm cov txheej txheem ntawm cov lej nyob rau hauv cov lej tiag. Yog tias a_n yog n^th lub sij hawm ntawm ib ntu, peb txhais cov kab ke los ntawm lossis los ntawm ib qho ₁, a ₂, …, a_n, …. Piv txwv li, xav txog ntu 1, ½, ⅓, …, ¹ / _{n,…. Nws tuaj yeem txhais tau tias yog {1/n}.}

Nws muaj peev xwm los txhais cov koob uas siv cov kab ke. Ib tug series yog cov sum ntawm cov nqe lus ntawm ib theem zuj zus. Yog li ntawd, rau txhua qhov sib lawv liag, muaj kev sib txuas ua ke thiab vice versa. Yog tias {a_{n} yog qhov sib lawv liag hauv kev txiav txim siab, ces, series tsim los ntawm cov kab ke tuaj yeem sawv cev li:}

Yog li, hauv qhov piv txwv saum toj no, cov kab sib txuas yog 1+¹/₂+^{1 /}3_{+ … +} 1^/ n _{+ ….}

Raws li cov npe qhia, lub hwj chim series yog ib hom tshwj xeeb ntawm series thiab nws tau siv dav hauv Kev Ntsuas Tus lej thiab lwm yam kev ua lej. Taylor series yog ib lub zog tshwj xeeb uas muab lwm txoj hauv kev thiab yooj yim-rau-siv txoj hauv kev uas sawv cev rau cov haujlwm zoo.

Power series yog dab tsi?

Lub hwj chim series yog ib daim ntawv

uas yog convergent (tejzaum) rau qee lub sijhawm nruab nrab ntawm c. Cov coefficients a_{ntuaj yeem yog tus lej tiag lossis nyuaj, thiab tsis muaj x; i.e. dummy variable.}

Piv txwv, los ntawm kev teeb tsa _n=1 rau txhua n, thiab c=0, lub zog series 1 + x + x ² +…..+ x^{+… yog tau. Nws yog ib qho yooj yim los soj ntsuam tias thaum x ε (-1, 1), lub hwj chim series converges rau 1/(1-x).}

Ib lub hwj huam sib hloov thaum x=c. Lwm qhov tseem ceeb ntawm x uas lub hwj chim series converges yuav ib txwm coj daim ntawv ntawm ib tug qhib lub sij hawm nruab nrab ntawm c. Ntawd yog, yuav muaj tus nqi 0≤ R ≤ ∞ xws li rau txhua x txaus siab |x-c|≤ R, lub hwj chim series yog convergent thiab rau txhua x txaus siab |x-c|> R, lub hwj chim series yog divergent. Qhov no tus nqi R yog hu ua lub vojvoog ntawm convergence ntawm lub hwj chim series (R tuaj yeem coj tus nqi tiag tiag lossis qhov zoo infinity).

Power series tuaj yeem muab ntxiv, rho tawm, sib npaug thiab faib siv cov cai hauv qab no. Xav txog ob lub hwj chim series:

Tam sim no,

i.e. zoo li cov ntsiab lus ntxiv los yog rho tawm ua ke. Tsis tas li ntawd, nws muaj peev xwm muab faib thiab faib ob lub hwj chim series siv tus kheej,

Taylor series yog dab tsi?

Taylor series yog txhais rau kev ua haujlwm f (x) uas yog qhov sib txawv tsis kawg ntawm lub sijhawm. Piv txwv tias f (x) yog qhov sib txawv ntawm lub sijhawm nruab nrab ntawm c. Ces lub hwj chim series uas yog muab los ntawm

yog hu ua Taylor series nthuav dav ntawm txoj haujlwm f (x) txog c. (Ntawm no f⁽ⁿ⁾ (c) denote the n^{th derivative at x=c). Hauv Kev Tshawb Fawb Txog Kev Tshawb Fawb, tus lej ntawm cov ntsiab lus nyob rau hauv qhov kev nthuav dav tsis kawg yog siv los suav cov txiaj ntsig ntawm cov ntsiab lus uas cov koob yog convergent rau qhov tseem ceeb muaj nuj nqi.}

A muaj nuj nqi f (x) tau hais tias yog kev tshuaj xyuas hauv lub sijhawm (a, b), yog tias rau txhua x ε (a, b), Taylor series ntawm f (x) converges rau qhov ua f (x). Piv txwv li, 1/(1-x) yog analytic ntawm (-1, 1), txij thaum nws Taylor expansion 1+x+x²+….+ x ^{+… converges rau lub luag haujlwm ntawm lub sijhawm ntawd, thiab e}x ^{yog analytic nyob txhua qhov chaw, txij li Taylor series ntawm e}x ^{converges rau e} x ^{rau txhua tus lej x.}

Qhov txawv ntawm Hwj chim series thiab Taylor series yog dab tsi?

1. Taylor series yog ib chav tshwj xeeb ntawm lub hwj chim series txhais tau tias tsuas yog rau cov dej num uas yog infinitely txawv ntawm ib co qhib lub sij hawm.

2. Taylor series coj daim ntawv tshwj xeeb

whereas, lub hwj chim series tuaj yeem yog ib qho ntawm daim ntawv