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Çàóâàæåííÿ. Ðîçêëàä ìíîãî÷ëåíà p n (x) çà ñòåïåíÿìè<br />
õ – õ 0 çä³éñíþºòüñÿ îäíîçíà÷íî.<br />
Öå ä³éñíî òàê, îñê³ëüêè êîåô³ö³ºíòè ðîçêëàäó âèçíà÷àþòüñÿ<br />
îäíîçíà÷íî ÷åðåç çíà÷åííÿ ìíîãî÷ëåíà p n (x) ³ éîãî<br />
ïîõ³äíèõ â òî÷ö³ õ = õ 0 .<br />
7.15.2. Ôîðìóëà á³íîìà Íüþòîíà<br />
Íåõàé ó ôîðìóë³ (7.15.6) ( ) = ( + ) , ∈ ,<br />
pn<br />
x a x n N à õ 0 =0.<br />
Òîä³ ç óðàõóâàííÿì ôîðìóëè (7.9.6) ìàòèìåìî<br />
Ïîçíà÷èìî<br />
( )<br />
( −1... ) ( − + 1)<br />
n n nn n k<br />
n−k k<br />
a+ x = ∑ a x . (7.15.7)<br />
k=<br />
0 k!<br />
( −1... ) ( − + 1)<br />
nn n k<br />
k!<br />
k<br />
n<br />
n<br />
= C . (7.15.8)<br />
k<br />
Ñèìâîë C ìຠãëèáîêèé çì³ñò ó êîìá³íàòîðèö³. Ïðî öå,<br />
n<br />
÷èòà÷, âè óçíàºòå ï³çí³øå, ïðè âèâ÷åíí³ òåî𳿠éìîâ³ðíîñòåé.<br />
Ôîðìóëà (7.15.7) ñïðàâåäëèâà äëÿ áóäü-ÿêîãî õ ä³éñíî¿ â³ñ³<br />
Îx. Çîêðåìà, ïðè x = b ç óðàõóâàííÿì (7.15.8) áóäåìî ìàòè<br />
( )<br />
n n<br />
k n−k k<br />
n<br />
k=<br />
1<br />
a+ b = ∑ C a b . (7.15.9)<br />
Öå ³ º çíàìåíèòà ôîðìóëà á³íîìà Íüþòîíà. Âîíà ÷àñòî<br />
çàñòîñîâóºòüñÿ ó òåîðåòè÷íèõ ³ ïðàêòè÷íèõ ïðîáëåìàõ ìàòåìàòèêè.<br />
Äî ðå÷³, ìè ¿¿ óæå çàñòîñîâóâàëè (äèâ. ï. 5.2.5).<br />
Çàóâàæåííÿ. Ñèìâîë<br />
C<br />
k<br />
n<br />
k<br />
C ìîæíà çîáðàçèòè ó âèãëÿä³:<br />
n<br />
n!<br />
=<br />
k! n− k !<br />
.<br />
( )<br />
×èòà÷åâ³ ïðîïîíóºìî öå òâåðäæåííÿ äîâåñòè ñàìîñò³éíî.<br />
7.15.3. Ôîðìóëà Òåéëîðà äëÿ ôóíêö³¿<br />
Íåõàé çàäàíà ôóíêö³ÿ ó = f(x), ùî íå º ìíîãî÷ëåíîì n-ãî<br />
ñòåïåíÿ ³ ÿêà ìຠïîõ³äí³ äî (n + 1)-ãî ïîðÿäêó âêëþ÷íî ó<br />
òî÷ö³ x 0 ³ ¿¿ δ-îêîë³.<br />
Ðîçãëÿíåìî çàäà÷ó ïðî íàáëèæåííÿ (àïðîêñèìàö³þ) ö³º¿<br />
ôóíêö³¿ ìíîãî÷ëåíîì. Ç ö³ºþ ìåòîþ ââåäåìî â ðîçãëÿä<br />
ìíîãî÷ëåí<br />
( k ) ( x0<br />
) ( )<br />
n f<br />
pn<br />
( x)<br />
= ∑ x −x<br />
k<br />
k=<br />
0 !<br />
0<br />
k<br />
. (7.15.10)<br />
Îñê³ëüêè çã³äíî ç çàóâàæåííÿì ïîïåðåäíüîãî ïóíêòó<br />
êîåô³ö³ºíòè ôîðìóëè Òåéëîðà âèçíà÷àþòüñÿ îäíîçíà÷íî, òî<br />
ïîâèíí³ âèêîíóâàòèñÿ òàê³ ñï³ââ³äíîøåííÿ<br />
( k )<br />
( )<br />
( k )<br />
n ( )<br />
f x0 = p x0 , k = 0,1,2,..., n. (7.15.11)<br />
Òàêèì ÷èíîì, ó â³äïîâ³äíîñò³ äî ñòðóêòóðè ìíîãî÷ëåíà<br />
(7.15.10) çíà÷åííÿ ôóíêö³¿ f(õ) ³ ¿¿ ïîõ³äíèõ äî n-ãî ïîðÿäêó<br />
âêëþ÷íî ñï³âïàäàþòü ç â³äïîâ³äíèìè çíà÷åííÿìè ìíîãî-<br />
÷ëåíà p n (x) ó ö³é òî÷ö³. Ïðè öüîìó ïîáóäîâàíèé ìíîãî÷ëåí<br />
çà ôîðìóëîþ (7.15.10) íàçèâàþòü ìíîãî÷ëåíîì Òåéëîðà äëÿ<br />
ôóíêö³¿ f(x).<br />
Äàë³ ââåäåìî ð³çíèöþ<br />
f(x) –p n (x) =R n (x). (7.15.12)<br />
Ïðè öüîìó ôóíêö³þ R n (x) íàçâåìî çàëèøêîâèì ÷ëåíîì.<br />
Î÷åâèäíî, ùî â³í çã³äíî ç ôîðìóëàìè (7.15.11) ìຠâëàñòèâîñò³<br />
′<br />
( n)<br />
R x = R x = ... = R ( x ) = 0 .<br />
( ) ( )<br />
n 0 n 0 n 0<br />
Äëÿ ç’ÿñóâàííÿ ñòðóêòóðè ôóíêö³¿ R n (x) ïîð³âíÿºìî ¿¿ ç<br />
n+<br />
ôóíêö³ºþ ϕ ( x) = ( x− x ) 1<br />
0<br />
(ÿñíî, ùî öÿ ôóíêö³ÿ â òî÷ö³ x = x 0<br />
ìຠò³ ñàì³ âëàñòèâîñò³, ùî ³ ôóíêö³ÿ R n (x)). Äëÿ ö³º¿ ìåòè<br />
ðîçãëÿíåìî:<br />
Rn<br />
( x)<br />
n+<br />
( x − x ) 1<br />
0<br />
.<br />
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