Cov tub ntxhais kawm ntawv qib siab yuav tsum paub tias qhov sib npaug ntawm qee lub hwj chim series koom nrog lub sijhawm ntawm kev sib koom ua ke ntawm cov koob muab hloov mus ua qhov txuas ntxiv thiab tsis txwv cov sij hawm sib txawv. Cov lus nug tshwm sim: nws puas tuaj yeem lees paub tias qhov kev ua haujlwm tsis txaus ntseeg f(x) yog cov lej ntawm qee lub zog? Qhov ntawd yog, nyob rau hauv cov xwm txheej twg tuaj yeem ua haujlwm f(x) sawv cev los ntawm lub hwj chim series? Qhov tseem ceeb ntawm lo lus nug no nyob rau hauv qhov tseeb hais tias nws muaj peev xwm yuav kwv yees li hloov lub function f(x) los ntawm cov sum ntawm thawj ob peb nqe lus ntawm lub hwj chim series, uas yog, los ntawm ib tug polynomial. Xws li kev hloov ntawm kev ua haujlwm los ntawm kev qhia yooj yim - polynomial - kuj yooj yim thaum daws qee qhov teeb meem ntawm kev txheeb xyuas lej, xws li: thaum daws qhov sib xyaw, thaum xam cov zauv sib txawv, thiab lwm yam.
Nws tau raug pov thawj tias rau qee qhov ua haujlwm f(х) qhov twg derivatives mus txog (n + 1)th xaj, suav nrog qhov kawg, tuaj yeem suav hauv zej zog (α - R; x0 + R) ntawm qee qhov taw tes x=α cov qauv siv tau:
Cov mis no muaj npe tom qab tus kws tshawb fawb nto moo Brook Taylor. Cov koob uas tau los ntawm yav dhau los yog hu ua Maclaurin series:
Txoj cai uas ua rau nws muaj peev xwm nthuav dav hauv Maclaurin series:
- Kev txiav txim siab ntawm thawj, thib ob, thib peb … xaj.
- Xaiv dab tsi cov derivatives ntawm x=0 yog sib npaug.
- Sau cov Maclaurin series rau txoj haujlwm no, thiab tom qab ntawd txiav txim siab lub sijhawm ntawm nws qhov kev sib koom ua ke.
- Txiav txim siab lub sijhawm (-R; R) qhov seem ntawm Maclaurin formula
R (x) -> 0 rau n -> infinity. Yog tias muaj, muaj nuj nqi f(x) hauv nws yuav tsum sib npaug nrog cov lej ntawm Maclaurin series.
Tam sim no xav txog Maclaurin series rau cov haujlwm ntawm tus kheej.
1. Yog li, thawj tus yuav yog f(x)=ex. Ntawm chav kawm, raws li nws cov yam ntxwv, xws li ib tug muaj nuj nqi muaj derivatives ntawm ntau yam kev txiav txim, thiab f(k)(x)=ex, qhov twg k sib npaug tag nrho. ntuj naj npawb. Cia peb hloov x=0. We get f(k)(0)=e0=1, k=1, 2… yuav zoo li no:
2. Maclaurin series rau qhov ua haujlwm f(x)=sin x. Tam sim ntawd qhia meej tias txoj haujlwm rau txhua qhov tsis paub yuav muaj cov txiaj ntsig, ntxiv rau f'(x)=cos x=sin(x+n/2), f '' (x)=-sin x=sin(x+2n/2)…, f(k)(x)=sin(x+k n/2), qhov twg k yog sib npaug rau txhua tus lej. Ntawd yog, tom qab ua cov lej yooj yim, peb tuaj yeem xaus lus tias koob rau f(x)=sin x yuav zoo li no:
3. Tam sim no cia sim xav txog qhov ua haujlwm f(x)=cos x. Nws yog rau txhua tus tsis paubmuaj derivatives ntawm arbitrary txiav txim, thiab |f(k)(x)|=|cos(x+kp/2)|<=1, k=1, 2… Ntxiv dua thiab, tom qab ua qee qhov kev suav, peb tau txais cov koob rau f(x)=cos x yuav zoo li no:
Yog li, peb tau teev cov haujlwm tseem ceeb tshaj plaws uas tuaj yeem nthuav dav hauv Maclaurin series, tab sis lawv tau ntxiv los ntawm Taylor series rau qee qhov haujlwm. Tam sim no peb yuav sau lawv. Nws tseem tsim nyog sau cia tias Taylor thiab Maclaurin series yog ib feem tseem ceeb ntawm kev coj ua ntawm kev daws teeb meem hauv kev ua lej siab dua. Yog li, Taylor series.
1. Thawj yuav yog ib tug series rau f-ii f(x)=ln(1+x). Raws li nyob rau hauv cov piv txwv yav dhau los, muab rau peb f (x)=ln (1 + x), peb tuaj yeem ntxiv cov koob siv daim ntawv dav dav ntawm Maclaurin series. Txawm li cas los xij, rau txoj haujlwm no, Maclaurin series tuaj yeem tau txais ntau dua yooj yim. Tom qab kev koom ua ke ib co geometric series, peb tau txais ib series rau f(x)=ln(1+x) ntawm cov qauv no:
2. Thiab qhov thib ob, uas yuav yog qhov kawg hauv peb tsab xov xwm, yuav yog cov koob rau f (x) u003d arctg x. Rau x nyob rau lub sijhawm [-1; 1], qhov nthuav dav siv tau:
Tseem. Kab lus no tau tshuaj xyuas cov feem ntau siv Taylor thiab Maclaurin series hauv kev ua lej siab dua, tshwj xeeb, hauv kev lag luam thiab kev kawm qib siab.
