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àáî<br />
dk<br />
= dt<br />
1 . (11.9.16)<br />
10 5 2<br />
k − k<br />
Ïðî³íòåãðóºìî ë³âó ÷àñòèíó ð³âíîñò³ (11.9.16):<br />
∫<br />
10 k<br />
⎡<br />
⎤<br />
dx<br />
=<br />
⎢ 1 ⎥<br />
= =<br />
− k dx = k dk 10 − x<br />
⎢<br />
⎣ 2<br />
⎥<br />
⎦<br />
1 2<br />
dk<br />
x = k<br />
⎢<br />
⎥<br />
1 1<br />
2<br />
−<br />
∫ 5<br />
5 2<br />
2<br />
5<br />
2ln 10 ln ln<br />
=− − x + C =<br />
C<br />
1<br />
5 2<br />
( 10 k )<br />
2<br />
− .<br />
ßêùî òåïåð ïðî³íòåãðóâàòè ïðàâó ÷àñòèíó ð³âíîñò³<br />
1<br />
(11.8.16), òî ç íå¿ îòðèìàºìî ( )<br />
2 10 5<br />
2<br />
k − = Ce −t<br />
. Öåé ðîçâ’ÿçîê<br />
— çàãàëüíèé. Âèêîðèñòîâóþ÷è ïî÷àòêîâó óìîâó, çíàéäåìî<br />
øóêàíó ôóíêö³þ (ñåðåäíþ ôîíäîîçáðîºí³ñòü ï³äïðèºìñò-<br />
5<br />
âà) ó âèãëÿä³ () ( ) 2<br />
ÂÏÐÀÂÈ<br />
k t = 99995e −t + 10 .<br />
11.46. Ðîçâ’ÿçàòè ð³âíÿííÿ (11.9.9) çà óìîâàìè:<br />
µ = 0.2, ρ = 0.8, Φ(0) = 10 6 ãðí.<br />
11.47. Ðîçâ’ÿçàòè ð³âíÿííÿ (11.9.14) çà óìîâàìè:<br />
−1<br />
( ) , 10 , 1.9, k(0) 10<br />
2<br />
qk<br />
2k<br />
= µ = ν = =<br />
k + 1<br />
.<br />
Òåìà 12<br />
Ðÿäè<br />
Ðÿäè ÿâëÿþòü ñîáîþ ïðîñòèé ³ äóæå äîñêîíàëèé ³íñòðóìåíò<br />
ìàòåìàòè÷íîãî àíàë³çó ÿê äëÿ òåîðåòè÷íèõ äîñë³äæåíü,<br />
òàê ³ äëÿ íàáëèæåíîãî îá÷èñëåííÿ çíà÷åíü ôóíêö³é é ïîáóäîâè<br />
íàáëèæåíèõ ðîçâ’ÿçê³â äèôåðåíö³àëüíèõ ð³âíÿíü.<br />
12.1. ×ÈÑËÎ<strong>²</strong> ÐßÄÈ<br />
12.1.1. Ïîíÿòòÿ ÷èñëîâîãî ðÿäó<br />
Ðîçãëÿíåìî ÷èñëîâó ïîñë³äîâí³ñòü {a n }. Ç’ºäíàâøè çíàêîì<br />
àëãåáðà¿÷íîãî äîäàâàííÿ ÷ëåíè ö³º¿ ïîñë³äîâíîñò³,<br />
îòðèìàºìî âèðàç, ùî ì³ñòèòü íåñê³í÷åííå ÷èñëî äîäàíê³â,<br />
öåé âèðàç ³ íàçèâàºòüñÿ ÷èñëîâèì ðÿäîì, àáî ïðîñòî ðÿäîì:<br />
a 1 + a 2 + a 3 + ... + a n + ... = ∑ a . (12.1.1)<br />
×èñëà a 1 , a 2 , ..., a n íàçèâàþòüñÿ ÷ëåíàìè ðÿäó, ÷ëåí a n ç<br />
äîâ³ëüíèì íîìåðîì — çàãàëüíèì ÷ëåíîì ðÿäó.<br />
Ââåäåìî ïîíÿòòÿ ÷àñòèííèõ (çð³çàíèõ) ñóì: S 1 = a 1 ,<br />
S 2 = a 1 + a 2 , S 3 = a 1 + a 2 + a 3 , ..., S n = a 1 + a 2 + a 3 + ... + a n . Îñê³ëüêè<br />
÷èñëî ÷ëåí³â ðÿäó íåñê³í÷åííå, òî ÷àñòèíí³ ñóìè<br />
ðÿäó óòâîðþþòü ïîñë³äîâí³ñòü ÷àñòèííèõ ñóì<br />
S 1 , S 2 , ..., S n , ... . (12.1.2)<br />
Îçíà÷åííÿ 12.1.1. ßêùî ïîñë³äîâí³ñòü ÷àñòèííèõ ñóì<br />
(12.1.2) ìຠñê³í÷åííó ãðàíèöþ lim S n<br />
= S , òî ðÿä íàçèâàºòüñÿ<br />
çá³æíèì, à S íàçèâàºòüñÿ éîãî ñóìîþ. ßêùî æ ïîñë³äî-<br />
n→∞<br />
âí³ñòü ÷àñòèííèõ ñóì (12.1.2) íå ìຠãðàíèö³ àáî âîíà<br />
äîð³âíþº íåñê³í÷åííîñò³, òî ðÿä íàçèâàºòüñÿ ðîçá³æíèì.<br />
Ïðèêëàä 12.1.1. Ïîêàæåìî, ùî ðÿä<br />
∞<br />
n=<br />
1<br />
n<br />
1 1 1 1 ∞ 1<br />
+ + + K+ + K= ∑<br />
1⋅2 2⋅3 3⋅4 n⋅ n + 1 n⋅ n + 1<br />
( ) n=<br />
1 ( )<br />
çá³ãàºòüñÿ. ³çüìåìî ñóìó S n ïåðøèõ n ÷ëåí³â ðÿäó<br />
450 451
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