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12.5.6. Äîâåäåííÿ ôîðìóëè Åéëåðà<br />
Âèêîðèñòîâóþ÷è ðîçêëàäàííÿ ó ðÿä Ìàêëîðåíà ôóíêö³é<br />
e x , sin x, cos x, îòðèìàºìî:<br />
2 3 4<br />
( ) ( ) ( ) ( )<br />
2 4 6<br />
ix ix ix ix ix ix ⎛ x x x ⎞<br />
e = 1+ + + + + K+ + K= ⎜1− + − + K⎟+<br />
1! 2! 3! 4! n! ⎝ 2! 4! 6! ⎠<br />
3 5 7<br />
⎛ x x x ⎞<br />
2<br />
+ i⎜x− + − + K ⎟ = cos x+ isin x ( i =−1)<br />
⎝ 3! 5! 7!<br />
. (12.5.12)<br />
⎠<br />
Òóò ³ — óÿâíà îäèíèöÿ (äèâ. äîäàòîê 1).<br />
Êð³ì ðîçãëÿíóòèõ ôóíêö³é e x , sin x, cos x, ó ðÿä Ìàêëîðåíà<br />
ìîæóòü áóòè ðîçêëàäåí³ ³ áàãàòî ³íøèõ ôóíêö³é.<br />
Çàì³ñòü ðÿäó Ìàêëîðåíà ìîæíà áóëî á ðîçãëÿíóòè ³<br />
á³ëüø çàãàëüíèé ðÿä Òåéëîðà (12.5.9). Âèêëàäåíå ö³ëêîì<br />
ïåðåíîñèòüñÿ ³ íà ðîçêëàäàííÿ ðîçãëÿíóòèõ ôóíêö³é ó ðÿä<br />
Òåéëîðà.<br />
Äëÿ ðîçêëàäàííÿ äåÿêèõ ôóíêö³é ó ðÿä ìîæíà âèêîðèñòîâóâàòè<br />
³íøó âëàñòèâ³ñòü ñòåïåíåâèõ ðÿä³â — ¿õ ïî÷ëåííå<br />
³íòåãðóâàííÿ.<br />
Ïðèêëàä 12.5.4. Ðîçêëàñòè çà äîïîìîãîþ ïî÷ëåííîãî ³íòåãðóâàííÿ<br />
äî ñòåïåíåâèõ ðÿä³â ôóíêö³¿ ln(1 + x) ³ arctg x.<br />
2 3<br />
n<br />
Ðîçãëÿíåìî ðÿä 1+ x+ x + x + K+ x + K. Äàíèé ðÿä º ãåîìåòðè÷íîþ<br />
ïðîãðåñ³ºþ ç³ çíàìåííèêîì q = x. Ïðè x < 1 ðÿä<br />
çá³ãàºòüñÿ ³ éîãî ñóìà äîð³âíþº<br />
1<br />
1<br />
2 3<br />
n<br />
1 x x x x<br />
− x = + + + + + +<br />
n<br />
K K. (12.5.13)<br />
1<br />
гâí³ñòü (12.5.13) º ðîçêëàäàííÿì ôóíêö³¿ f( x)<br />
= â<br />
1 − x<br />
ñòåïåíåâèé ðÿä. ϳäñòàâëÿþ÷è â íüîãî –t çàì³ñòü x, îòðèìà-<br />
ºìî ð³âí³ñòü<br />
1<br />
2 3<br />
n n<br />
1 t t t ( 1)<br />
t<br />
1+ t = − + − + K + − + K,<br />
ÿêà ñïðàâåäëèâà ïðè t < 1. dzíòåãðóºìî öåé ñòåïåíåâèé ðÿä<br />
ïî÷ëåííî ó ìåæàõ â³ä 0 äî x ( x < 1 ). Ìàºìî<br />
n n<br />
( K)<br />
x<br />
dt<br />
x<br />
x<br />
2 3<br />
∫ ln ( 1 t) ln ( 1 x) 1 t t t ( 1)<br />
t dt<br />
01<br />
t = + 0<br />
= + = ∫ − + − + K<br />
+<br />
+ − + =<br />
0<br />
Çâ³äñè<br />
2 3 4 n+<br />
1<br />
x t x t x t x n t x<br />
= t − + − + K+ ( − 1)<br />
+ K=<br />
0 2 0 3 0 4 0 n + 1 0<br />
2 3 4 n+<br />
1<br />
x x x n x<br />
= x − + − + K+ ( − 1)<br />
+ K.<br />
2 3 4 n + 1<br />
2 3 4 n+<br />
1<br />
x x x n x<br />
ln( 1+ x) = x− + − + K+ ( − 1)<br />
+ K. (12.5.14)<br />
2 3 4 n + 1<br />
Ïîêàæåìî, ùî îòðèìàíå ðîçêëàäàííÿ ôóíêö³¿ ln(1 + x) ó<br />
ñòåïåíåâèé ðÿä ñïðàâåäëèâå ³ ïðè x = 1. ijéñíî ïðè x =1<br />
ë³âà ÷àñòèíà (12.4.14) äîð³âíþº ln 2, à ïðàâà ÷àñòèíà —<br />
çá³æíèé çà îçíàêîþ Ëåéáí³öà ÷èñëîâèé ðÿä<br />
1 1 1 n−<br />
1− + − + K+ ( − 1) 1 1 + K. (12.5.15)<br />
2 3 4<br />
n<br />
Çàëèøàºòüñÿ ïåðåâ³ðèòè ñïðàâåäëèâ³ñòü ð³âíîñò³<br />
1 1 1 n−<br />
ln 2 = 1− + − + K+ ( − 1) 1 1 + K. (12.5.16)<br />
2 3 4<br />
n<br />
Äëÿ öüîãî ç³íòåãðóºìî â³ä 0 äî 1 âèðàç<br />
n<br />
1<br />
2 3 n−1<br />
n−1<br />
n t<br />
= 1− t+ t − t + K + ( − 1) t + ( −1)<br />
,<br />
1+ t<br />
1+<br />
t<br />
îòðèìàíèé ó ðåçóëüòàò³ ä³ëåííÿ îäèíèö³ íà 1 + t. Ìàºìî<br />
1 1<br />
n<br />
dt<br />
1 ⎛<br />
1<br />
( ) 2 3 n−<br />
n−<br />
ln 1 ln 2 1 ( 1) 1 n t ⎞<br />
∫ = + t = = ∫⎜ − t+ t − t + K+ − t + ( − 1)<br />
⎟dt<br />
=<br />
01+ t 0<br />
0⎝<br />
1+<br />
t⎠<br />
472 473
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