Maclaurin thiab decomposition ntawm ib co zog

Kawm tshaj kev kawm txog zauv yuav tsum paub hais tias lub sum ntawm ib lub hwj chim series nyob rau hauv lub luv ntawm convergence ntawm ib tug xov tooj ntawm peb, yog ib tug tas mus li thiab unlimited tooj ntawm lub sij hawm ib tug differentiated muaj nuj nqi. Cov lus nug tshwm sim: yog nws tau mus hais lus cam uas muab ib tug arbitrary muaj nuj nqi f (x) - yog tus sum ntawm ib lub hwj chim series? Uas yog, nyob rau hauv dab tsi tej yam kev mob lub f-tej f (x) yuav sawv cev los ntawm ib lub hwj chim series? Qhov tseem ceeb ntawm qhov teeb meem no yog hais tias nws yog tau los hloov kwv yees li £ Theological f (x) yog lub sum ntawm thawj ob peb cov ntsiab lus ntawm ib lub hwj chim series, uas yog ib tug polynomial. Xws li ib tug hloov nuj nqi yog heev yooj yim qhia - polynomial - yog yooj yim thiab nyob rau hauv kev daws tej yam teeb meem nyob rau hauv lej tsom xam, namely nyob rau hauv kev daws integrals thaum xam differential sib npaug , etc ...

Nws yog muaj pov thawj, uas rau ib co f-ii f (x), nyob rau lub derivatives ntawm (n + 1) -th thiaj yuav xam, xws li qhov tseeb nyob rau hauv lub cheeb tsam ntawm (α - R; x ₀ + R) ntawm ib tug taw tes x = α ncaj ncees mis yog:

Qhov no mis yog muaj npe tom qab tus naas ej tus paub txog Brooke Taylor. Ib tug xov tooj ntawm cov uas yog derived ntawm lub yav dhau los ib tug, yog hu ua ib tug Maclaurin series:

Ib tug txoj cai uas yuav ua rau nws tau mus tsim expansion nyob rau hauv ib tug Maclaurin series:

1. Txiav txim seb derivatives ntawm ib, ob, thib peb, ... kev txiav txim.
2. Xam dab tsi yog derivatives ntawm x = 0.
3. Cov ntaub ntawv Maclaurin series rau no muaj nuj nqi, thiab ces los mus txiav txim hauv lub luv ntawm convergence.
4. Txiav txim seb luv (-R; R), qhov twg residual ib feem ntawm mis Maclaurin

R _n (x) -> 0 rau n -> infinity. Yog hais tias ib tug tshwm sim, nws muaj nuj nqi f (x) yuav tsum yog ncaj rau tus sum ntawm cov Maclaurin series.

Xav txog tam sim no lub Maclaurin series rau tus neeg khiav dej num.

1. Yog li, tus thawj yuav tsum tau f (x) = e ^x. Ntawm cov hoob kawm, uas lawv cov yam ntxwv li f-Ia tau muab ib tug ntau yam ntawm kev txiav txim, thiab f ^{(k) (x)} = e ^x, qhov twg k yog sib npaug zos rau tag nrho cov lub ntuj tus xov tooj. Tus seev cev ntawm x = 0. Peb tau f ^{(k) (0)} = e ⁰ = 1, k = 1,2 ... Raws li cov foregoing, ib tug xov tooj ntawm e ^x Nws yuav ua tau raws li nram no:

2. Maclaurin series rau cov nuj nqi f (x) = txhaum x. Tam sim ntawd qhia kom meej hais tias f-tej rau tag nrho cov tsis paub hais tias derivatives yuav muaj, dua li f ^'(x) = cos x = kev txhaum (x + n / 2), ^{f' '(x)} = -sin x = kev txhaum (x + 2 * n / ²⁾ ..., f ^{(k) (x)} = kev txhaum (x + n * k / 2), qhov chaw uas k yog sib npaug zos rau tej zoo integer. Hais tias yog, ua tej yam yooj yim suav, peb yuav xaus uas lub series rau f (x) = txhaum x yuav zoo li no:

3. Tam sim no cia peb xav txog iju f-f (x) = cos x. Nws yog tsis paub hais tias rau tag nrho cov derivatives ntawm arbitrary kev txiav txim, thiab ^{| f (k) (x)} | = | Cos (x + k * n / 2) | <= 1, k = 1,2 ... Ib zaug ntxiv, nws tau ua ib co lus teb, peb pom tias lub series rau f (x) = cos x yuav zoo li no:

Yog li ntawd, peb tau teev cov tseem ceeb tshaj plaws nta uas yuav tsum txhab nyob rau hauv ib tug Maclaurin series, tab sis lawv Nkij los txuam rau cov Taylor series rau ib co zog. Tam sim no peb yuav sau lawv zoo li. Nws yuav tsum tau muab sau tseg tias Taylor series thiab Maclaurin series yog ib qho tseem ceeb ib feem ntawm qhov kev series ntawm kev txiav txim siab nyob rau hauv kev kawm leb siab. Yog li ntawd, Taylor series.

1. Tus thawj yog ib tug series ntawm f-ii f (x) = ln (1 + x). Raws li nyob rau hauv lub yav dhau los piv txwv li no, peb f (x) = ln (1 + x) yuav daim ntau quav tso ib tug xov tooj, siv cov kev daim ntawv ntawm Maclaurin series. tab sis rau no feature Maclaurin yuav tau yooj yim dua. Ua ke ib geometric series, peb tau ib tug xov tooj rau f (x) = ln (1 + x) ntawm tus qauv:

2. Thiab lub thib ob, uas yuav tsum yog qhov kawg nyob rau hauv no tsab xov xwm, yuav yog ib series rau f (x) = arctg x. Rau x teej tug mus rau hauv lub luv [-1; 1] yog siv tau decomposition:

Hais tias yog tag nrho. Nyob rau hauv no tsab xov xwm kuv tau surveyed lub feem ntau siv Taylor series thiab Maclaurin series nyob rau hauv kev kawm leb siab, tshwj xeeb yog nyob rau hauv lub economic thiab kev kawm ntawv qib siab.