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ÒÅÌÀ 5<br />
ÃÐÀÍÈÖß ×ÈÑËÎÂÈÕ ÏÎÑ˲ÄÎÂÍÎÑÒÅÉ<br />
5.1. ×ÈÑËÎ<strong>²</strong> ÏÎÑ˲ÄÎÂÍÎÑÒ² ÒÀ<br />
ÀÐÈÔÌÅÒÈ×Ͳ IJ¯ ÍÀÄ ÍÈÌÈ<br />
5.1.1. Îçíà÷åííÿ ³ ïðèêëàäè<br />
ßêùî êîæíîìó íàòóðàëüíîìó ÷èñëó n ³ç ìíîæèíè íàòóðàëüíèõ<br />
÷èñåë 1,2,3,…, n, …ïîñòàâëåíî ó â³äïîâ³äí³ñòü ïåâíå<br />
ä³éñíå ÷èñëî x n , òî ìíîæèíà ä³éñíèõ ÷èñåë<br />
x 1 , x 2 , x 3 , ... x n , ... (5.1.1)<br />
íàçèâàºòüñÿ ÷èñëîâîþ ïîñë³äîâí³ñòþ, àáî ïðîñòî ïîñë³äîâí³ñòþ.<br />
×èñëà x 1 , x 2 , x 3 , ... x n ,... áóäåìî íàçèâàòè åëåìåíòàìè<br />
(àáî ÷ëåíàìè) ïîñë³äîâíîñò³ (5.1.1), ñèìâîë x n º çàãàëüíèé<br />
åëåìåíò (àáî ÷ëåí) ö³º¿ ïîñë³äîâíîñò³, à ÷èñëî n º éîãî<br />
íîìåð. Ö³ëêîì î÷åâèäíî, ùî ïîñë³äîâí³ñòü º ìíîæèíîþ íåñê³í÷åííîþ<br />
³ âñ³ åëåìåíòè ïîñë³äîâíîñò³ (5.1.1) ð³çí³, ïðèíàéìí³<br />
âîíè â³äð³çíÿþòüñÿ ì³æ ñîáîþ ñâî¿ìè íîìåðàìè.<br />
Ñêîðî÷åíî ïîñë³äîâí³ñòü (5.1.1) çàïèñóþòü òàê: {x n }.<br />
Çàóâàæåííÿ. Ó â³äïîâ³äíîñò³ äî îçíà÷åííÿ ôóíêö³¿,<br />
ïîñë³äîâí³ñòü º ôóíêö³ÿ íàòóðàëüíîãî àðãóìåíòó:<br />
x n =f(n), n∈N.<br />
Ïîñë³äîâíîñò³ ìîæóòü áóòè çàäàí³ ïî-ð³çíîìó. Ïîêàæåìî<br />
öå íà ïðèêëàäàõ.<br />
1<br />
Ïðèêëàä 5.1.1. x n<br />
= (ïîñë³äîâí³ñòü çàäàíà àíàë³òè÷íèì<br />
n<br />
ñïîñîáîì).<br />
Ïðèêëàä 5.1.2. p n =2Rn ⋅ sin(π/n), R — äîäàòíå ÷èñëî (ïîñë³äîâí³ñòü<br />
p n çàäàíà àíàë³òè÷íèì ñïîñîáîì).<br />
Ïðèêëàä 5.1.3. s n =1/2R 2 n ⋅ sin(2π/n), R — äîäàòíå ÷èñëî<br />
(ïîñë³äîâí³ñòü s n çàäàíà àíàë³òè÷íèì ñïîñîáîì).<br />
Ïðèêëàä 5.1.4. x n+1 =x n +d, d — ñòàëå ÷èñëî (ðåêóðåíòíèé<br />
Ïðèêëàä 5.1.5. x n+1 = x n q, q — ñòàëå ÷èñëî (ðåêóðåíòíèé<br />
ñïîñ³á çàäàííÿ ïîñë³äîâíîñò³).<br />
Ïðèêëàä 5.1.6 x 1 =0,3; x 2 =0,33; x 3 =0,333; ... º îïèñóþ÷èé<br />
ñïîñ³á óòâîðåííÿ ïîñë³äîâíîñò³.<br />
Ïðèêëàä 5.1.7. Ïîñë³äîâí³ñòü Ô³áîíà÷÷³ 1 :<br />
x 1 = 1, x 2 = 1, x n = x n-1 + x n-2 (n∈N, n ≥ 3).<br />
Öÿ ïîñë³äîâí³ñòü çàäàíà òåæ ðåêóðåíòíèì ñïîñîáîì, àëå<br />
á³ëüø ñêëàäí³øå, í³æ ó ïðèêëàä³ 5.1.4.<br />
5.1.2. Àðèôìåòè÷í³ îïåðàö³¿ íàä ïîñë³äîâíîñòÿìè<br />
Íàä ïîñë³äîâíîñòÿìè ìîæíà ââåñòè àðèôìåòè÷í³ îïåðàö³¿.<br />
Íåõàé çàäàí³ äâ³ ïîñë³äîâíîñò³ {x n } ³ {y n }. Òîä³ ñóìîþ, ð³çíèöåþ,<br />
äîáóòêîì ³ ÷àñòêîþ íàçèâàþòü â³äïîâ³äíî ïîñë³äîâíîñò³<br />
{s n }, {q n }, {m n } ³ {r n }, ÷ëåíè ÿêèõ îá÷èñëþþòü çà ïðàâèëàìè:<br />
xn<br />
sn = xn + yn, qn = xn − yn, mn = xn ⋅ yn, rn<br />
= , n∈N,<br />
y<br />
ïðè÷îìó ÷àñòêà âèçíà÷åíà ò³ëüêè äëÿ òèõ ïîñë³äîâíîñòåé<br />
{y n }, â ÿêèõ æîäåí ÷ëåí íå äîð³âíþº íóëþ.<br />
5.1.3. Îáìåæåí³ òà íåîáìåæåí³ ïîñë³äîâíîñò³<br />
Îçíà÷åííÿ 5.1.1. Ïîñë³äîâí³ñòü {x n } íàçèâàþòü îáìåæåíîþ<br />
çâåðõó (çíèçó), ÿêùî ³ñíóº òàêå ÷èñëî Ì (m), ùî äëÿ<br />
âñ³õ íàòóðàëüíèõ ÷èñåë n âèêîíóºòüñÿ íåð³âí³ñòü<br />
x n ≤ M (x n ≥ m).<br />
Ïðèêëàä 5.1.8. Ïîñë³äîâí³ñòü {n} îáìåæåíà çíèçó ÷èñëîì<br />
1.<br />
Ïðèêëàä 5.1.9. Ïîñë³äîâí³ñòü {-n} îáìåæåíà çâåðõó ÷èñëîì<br />
−1.<br />
Îçíà÷åííÿ 5.1.2. Ïîñë³äîâí³ñòü {x n } íàçèâàþòü îáìåæåíîþ,<br />
ÿêùî ³ñíóþòü òàê³ ÷èñëà Ì ³ m, ùî äëÿ âñ³õ íàòóðàëüíèõ<br />
÷èñåë n âèêîíóºòüñÿ íåð³âí³ñòü<br />
m ≤ x n ≤ M.<br />
n<br />
ñïîñ³á çàäàííÿ ïîñë³äîâíîñò³). 1<br />
Ëåîíàðäî ϳçàíñüêèé (Ô³áîíà÷÷³) (áëèçüêî 1170 – ï³ñëÿ 1228) —<br />
³òàë³éñüêèé ìàòåìàòèê<br />
134 135
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