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 òàê³é çàãàëüí³é ïîñòàíîâö³ çàäà÷ó âàæêî ðîçâ’ÿçóâàòè.<br />
Òîìó côîðìóëüîâàíó çàäà÷ó ñïî÷àòêó áóäåìî ðîçâ’ÿçóâàòè<br />
íàáëèæåíî. ³çüìåìî ÷èñëî t ³ ðîç³á’ºìî éîãî íà n ÷àñòèí.<br />
Áóäåìî òåïåð ââàæàòè, ùî çà ïðîì³æîê t çä³éñíþºòüñÿ íàðàõóâàííÿ<br />
â³äïîâ³äíèõ r %. Òîä³ çà öèì ïðèïóùåííÿì áóäåìî ìàòè<br />
n<br />
íàáëèæåí³ çíà÷åííÿ íàêîïè÷åíî¿ ñóìè çà ïðîì³æêè ÷àñó<br />
t 2t 3t nt<br />
, , , ... = t .<br />
n n n n<br />
Ñïî÷àòêó çíàéäåìî Q 1 :<br />
t<br />
(1 kt<br />
Q = )<br />
1<br />
Q + 0<br />
kQ0 Q0 n<br />
= + n<br />
. Òóò<br />
r<br />
k = .<br />
100<br />
Äàë³ ïîñë³äîâíî çíàõîäèìî:<br />
2<br />
kt ⎛ kt ⎞ ⎛ kt ⎞<br />
Q2 = Q1 + Q1 = Q1 1 Q0<br />
1<br />
n<br />
⎜ + = +<br />
n<br />
⎟ ⎜<br />
n<br />
⎟<br />
⎝ ⎠ ⎝ ⎠ ,<br />
... ... ... ... ... ... ... ... ... ...<br />
kt<br />
n<br />
⎛<br />
n n 1 n 1 n 1<br />
1 kt ⎞ ⎛<br />
Q Q Q Q Q0<br />
1<br />
kt ⎞<br />
=<br />
−<br />
+<br />
−<br />
=<br />
−<br />
n<br />
⎜ + = +<br />
n<br />
⎟ ⎜<br />
n<br />
⎟<br />
⎝ ⎠ ⎝ ⎠ .<br />
Äëÿ òîãî ùîá çíàéòè òî÷íî ñóìó çà ïåðâèííîþ óìîâîþ<br />
çàäà÷³ òðåáà çä³éñíèòè ãðàíè÷íèé ïåðåõ³ä. Ïîçíà÷àþ÷è ÷åðåç<br />
Q(t) íàêîïè÷åíó ñóìó â ìîìåíò ÷àñó t áóäåìî ìàòè<br />
kt<br />
Qt () = lim Q<br />
0(1 ) n<br />
n<br />
= Q + . (5.2.3)<br />
n→∞<br />
n<br />
Öþ ãðàíèöþ ìîæíà áóäå ëåãêî çíàéòè, ÿêùî ìè áóäåìî<br />
n<br />
⎛ 1 ⎞<br />
çíàòè ãðàíèöþ ïîñë³äîâíîñò³ x n<br />
= ⎜1+<br />
⎟ . Òàê ìè ïðèõîäèìî<br />
⎝ n ⎠<br />
äî ïîíÿòòÿ ÷èñëà e, ââåäåíîãî Åéëåðîì 1 .<br />
1<br />
Åéëåð Ëåîíàðä (1707 – 1783) — âèäàòíèé øâåéöàðñüêèé ìàòåìàòèê,<br />
ìåõàí³ê, àñòðîíîì, ÷ëåí Ïåòåðáóðçüêî¿ àêàäå쳿 íàóê, á³ëüøó ÷àñòèíó<br />
æèòòÿ ïðîâ³â ó Ðîñ³¿.<br />
5.2.5. ×èñëî å<br />
⎛ 1 ⎞<br />
Ðîçãëÿíåìî ïîñë³äîâí³ñòü x n<br />
= ⎜1+<br />
⎟ . Ïîêàæåìî, ùî âîíà<br />
⎝ n ⎠<br />
çá³ãàºòüñÿ. Äëÿ öüîãî äîñòàòíüî äîâåñòè, ùî ïîñë³äîâí³ñòü<br />
{x n } — çðîñòàþ÷à ³ îáìåæåíà. Çàñòîñîâóþ÷è ôîðìóëó á³íîìà<br />
Íüþòîíà 1 (âîíà áóäå íàâåäåíà â íàñòóïíèõ ëåêö³ÿõ), îòðèìà-<br />
ºìî<br />
1 nn ( −1) 1 nn ( −1)( n−2) 1<br />
xn<br />
= 1 + n + + + ...<br />
n<br />
2 3<br />
2! n 3! n<br />
nn ( −1)( n−2)...[ n−( n−1)]. 1<br />
... +<br />
n<br />
n!<br />
n . (5.2.4)<br />
Òóò ìè âèêîðèñòàëè ñèìâîë n! (÷èòàþòü: “åí ôàêòîð³àë”),<br />
ÿêèé îçíà÷àº:<br />
n! =1⋅2⋅3 ... (n−1) n.<br />
Âèðàç (5.2.4) çîáðàçèìî â òàê³é ôîðì³:<br />
x n<br />
1 1 1 1 2<br />
= 2 + (1 − ) + (1 − )(1 − ) + ...<br />
2! n 3! n n<br />
1 1 2 1<br />
... + (1 − )(1 − )...(1 − n − ) . (5.2.5)<br />
n!<br />
n n n<br />
Çàóâàæèìî, ùî ³ç ð³âíîñò³ (5.2.4) áåçïîñåðåäíüî âèïëèâàº<br />
íåð³âí³ñòü x n ≥ 2.<br />
Çíàéäåìî òåïåð x n+1 :<br />
= 1 1 1 1 2<br />
2 + (1 ) (1 )(1 ) ...<br />
2! − +<br />
n+ 1 + 3! − n+ 1 − n+<br />
1<br />
+<br />
x n 1<br />
1 1 2 n<br />
... + (1 − )(1 − )...(1 − ).<br />
( n+ 1)! n+ 1 n+ 1 n+<br />
1<br />
n<br />
(5.2.6)<br />
1<br />
²ñààê Íüþòîí (1643 – 1727) — âåëèêèé àíãë³éñüêèé ô³çèê, ìåõàí³ê,<br />
àñòðîíîì ³ ìàòåìàòèê.<br />
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