Учебно-методическое пособие (1021371), страница 3

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. . · (3n)P130arcsin4n (n + 1)!2nn=1ÌåäÒðÓàÌ ÂÌÈ-2ÝÀ25∞PÇàäà÷à 2Íàéòè ðàäèóñ è èíòåðâàë ñõîäèìîñòè ñòåïåííîãî ðÿäà. Èññëåäîâàòü ñõîäèìîñòü ðÿäà íà êîíöàõ èíòåðâàëà.1∞Pn=132n + 1 nxn(n + 1)n=1∞ (−1)n+1 x3nPpn(n + 1)n=1∞P1(x − 1)n sinn+1n=1∞P(x + 10)nnp(−1)n(n + 1)n=1n∞Pn (x + 11)(−1)n2n ln nn=2n + 2∞Pln(x + 2)nnn=1(−1)n246791113Êà5∞Pxnn+181012142n xnpn(n2 + 1)n=1∞ 2n nPxnn=1 3n + 2∞P3n + 1(x + 1)n(−1)n−1n(n + 1)n=2∞ 2n (x − 3)nP√nn=1∞ ln(n + 1)P(x + 7)nn+1n=1∞ 3n (x − 2)2nPn ln nn=2∞Pn(x − 2)3n√n3 + 2n − 1n=1∞P32171921232527291(2x − 1) tgnn=1∞ ln nP√ (x + 1)nn=2 n n∞P(−1)nn√(x − 2)n34n + 2nn=1 n + 3 x + 1 n∞Plnn+12n=1∞Pn n xnnn=1 (n + 1)∞P2nxnnnn=1 2 + 3π n∞ xnP√ cos2nn=1∞ (x + 2)nPnnn=1 2 + 3n∞P16182022(−1)n (xn=1∞Parctg− 2)2nn2nn(x + 3)nnn=1 2 (5n − 2)∞P1(3x + 2)n sinn+2n=1∞ (x − 2)nPn=1 ln(n + 2)∞√P√ n + 1 − n (x − 2)nÌåäÒðÓàÌ ÂÌÈ-2ÝÀ15∞P24n=0262830∞P1(x − 2)3n arcsin √nn=1∞P1n(2x − 1)n arctg 2n +1n=1∞√Pn − n2 + 1 x3nn=1Çàäà÷à 3ÊàÏî ïðèçíàêó Âåéåðøòðàññà äîêàçàòü ðàâíîìåðíóþ ñõîäèìîñòüóíêöèîíàëüíîãî ðÿäà íà óêàçàííîì ïðîìåæóòêå.123n21 nn=1 (x + n )∞P1 1n(−1) 1 −n xnn=1∞ n + (−1)nPxsin √nn=2 n(n − 1)∞Px ∈ [2, 3]x ∈ [−3, −2]x ∈ [−2, 2]33∞P4n=2∞P5789√2e−(1−x n)n=1√n∞P n3 2 + sin(nx)3nn=1∞Pn2 xnnn=1 2 + x + 1√∞Px + 1 cos(nx)√3n5 + 1n=1∞Pxn!n=12101112∞ (n!) (x − 3)nP2n2n=1∞Pxn2nn=1 1 + xÊ141516172∞ (−1)n+1 tgn xPn(n + 1)n=1∞ P(−1)n+1 n−1xx+nn=1à13x ∈ [0, 1]x ∈ [1, 2]x ∈ (−∞, +∞)x ∈ [0, 5]ÌåäÒðÓàÌ ÂÌÈ-2ÝÀ6xn ln 1 + 3n2∞ 3nPn2n=1 x∞ (−1)n + 1Pn2nn=1 (x − 1)∞ 1Px n+nn=1 x∞ (x + 1) sin2 (nx)P√n n+1n=1x ∈ [0, 2]1 1x∈ − ,2 2x ∈ [2, 3]x ∈ [−3, −2]h π πix∈ − ,6 61 1x∈ − ,2 2x ∈ [4, 5]x ∈ [−2, −1]x ∈ [3, 5]x ∈ [−3, 0]3418192223242526n=1∞Pπxnnnsin nn!2n=1∞Pn3 xnnn=1 3 + x + 2∞Px2n√3n=1 n n + x∞Px3 ln 1 +2nlnnn=2√3∞Px2 + 4 sin nx√n n4 + 1n=1∞ (4 − x) cos2 nxP√n8 + 1n=1à27Ê282930x ∈ [1, 2]x ∈ [0, 3]x ∈ [0, 1]3x ∈ ,32ÌåäÒðÓàÌ ÂÌÈ-2ÝÀ2021nn+1n2n=1 (x − 4)∞Px+1 ln 1 +n ln2 (n + 1)n=1∞Px ln nsinnx−nn=2∞ 2 + (−1)nPx2nn=1∞ (π − x) cos2 (nx)P√4n7 + 1n=1∞Pn2nn=1 ln (x − 1)∞√Pn2n x + 1 e− x∞Px ∈ [0, π]x ∈ [4, 5]1x ∈ ,221x ∈ 0,2x ∈ [0, 4]x ∈ [0, 1]x ∈ [0, 2]x ∈ [0, 2]x ∈ [0, 4]Çàäà÷à 4Èñïîëüçóÿ ðàçëîæåíèå ýëåìåíòàðíûõ óíêöèé, ïîëó÷èòü ðàç-35ëîæåíèÿ äàííûõ óíêöèé â ñòåïåííûå ðÿäû ïî ñòåïåíÿì ïåðåìåííîé (x − x0 ).

Óêàçàòü îáëàñòè ñõîäèìîñòè ïîëó÷åííûõ ðÿäîâ.y = e2x1x0 = −12x2 + x − 6x0 = 03y = sin x cos xx0 = 04y = ln(3 − 2x)x0 = −45y = (1 − x) e−3xx0 = 1y=ÌåäÒðÓàÌ ÂÌÈ-2ÝÀ26x0 = 1y = x cos 3xx0 = 08y = ln(2x2 + 3x − 2)x0 = 19y = cos x cos 5xx0 = 0à710Ê2x − 1(x − 2)(x − 3)y=11y=2x − 32 − 5x − 3x2y = e−x2+6xx0 = −1x0 = 312y = sin2 xx0 = 013y = ln(x2 + 5x + 6)x0 = 214y=3x + 12x2 − 5x − 3x0 = 03615y = sin 2x cos 2xx0 = 016y=√8 − x2 + 2xx0 = 117 y = (x − 1) ln(2x − 3)5 − 2xx2 − 2x − 8xy = cos22y=x0 = 1ÌåäÒðÓàÌ ÂÌÈ-2ÝÀ18x0 = 2192x0 = −120y = (x − 2) e−x+4xx0 = 221y = ln(6x − 5 − x2 )x0 = 322y = sin 3x sin xx0 = 022324x0 = 0x0 = 0x2y=(x + 1)(x + 2)x0 = 126y = cos x sin2 xx0 = 027y = (3 − x) e3x−1x0 = 3281y=√x2 − 6x + 18x0 = 3à25Ê1 − e−xy=2rx1 + 2xy = ln 31−x292 − x2y = ln √1 + x2x0 = 03730y=x+3(x + 1)(x − 2)x0 = 0Çàäà÷à 5ÌåäÒðÓàÌ ÂÌÈ-2ÝÀà) àçëîæèòü óíêöèþ y = f (x), çàäàííóþ íà ïîëóïåðèîäå(0, l), â ðÿä Ôóðüå ïî êîñèíóñàì.

Ïîñòðîèòü ãðàèêè 2-îé, 3-åé÷àñòè÷íûõ ñóìì. Çàïèñàòü ðàâåíñòâî Ïàðñåâàëÿ äëÿ ïîëó÷åííîãîðÿäà.á) àçëîæèòü óíêöèþ y = f (x), çàäàííóþ íà ïîëóïåðèîäå(0, l), â ðÿä Ôóðüå ïî ñèíóñàì. Ïîñòðîèòü ãðàèêè 2-îé, 3-åé ÷àñòè÷íûõ ñóìì.â) àçëîæèòü óíêöèþ y = f (x) â ðÿä Ôóðüå, ïðîäîëæàÿ åå íàïîëóïåðèîäå (−l, 0) óíêöèåé, ðàâíîé íóëþ. Ïîñòðîèòü ãðàèêèâòîðîé, ÷åòâåðòîé ÷àñòè÷íûõ ñóìì.1 y = 1 − 3x l = 12 y = 4x + 4 l = 23 y = 2x + 1 l = 345 y = 4 − 4x l = 26 y = 3x + 2 l = 39 y = 4x − 1 l = 410 y = 2 − 3x l = 3à7 y = 4 − 2x l = 1Ê11 y = 4x − 3 l = 2y =1−xl=48 y = 1 − 2x l = 412y =x+1l=113 y = 2x − 1 l = 214 y = 1 − 4x l = 117 y = 3x − 1 l = 118 y = 3x + 3 l = 221 y = 3x + 1 l = 222 y = 2 − 4x l = 115y =x−1l=416 y = 4x + 1 l = 219 y = 2 − 2x l = 220 y = 3 − 3x l = 123 y = 2x + 2 l = 424 y = 4x + 3 l = 125 y = 2x − 2 l = 326 y = 3x − 2 l = 23827 y = 4x + 2 l = 129 y = 3 − 4x l = 128 y = 4x − 2 l = 230 y = 3x − 3 l = 2Çàäà÷à 6ÌåäÒðÓàÌ ÂÌÈ-2ÝÀÏðåäñòàâèòü èíòåãðàëîì Ôóðüå óíêöèþ, çàäàííóþ íà èíòåðâàëå (0, +∞), ïðîäîëæèâ åå íà âñþ ÷èñëîâóþ îñü ÷åòíûì è íå÷åòíûì îáðàçîì.1 y3 y55−7x,0<x≤7= 50,x>777−2x,0<x≤2= 70,x>244−7x,0<x≤7= 40,x>788−3x,0<x≤3= 80,x>32 2 − 9x, 0 < x ≤9= 20,x>9246Êà5 y7 y9 y81022−5x,0<x≤5y=20x>533−5x,0<x≤5y=30x>555−6x,0<x≤6y=50x>666−5x,0<x≤5y=60x>57 7 − 4x, 0 < x ≤4y=70x>43913151712141618202123Êà1933−8x,0<x≤8y= 30,x>88 8 − 5x, 0 < x ≤5y= 80,x>522−7x,0<x≤7y= 20,x>73 3 − 2x, 0 ≤ x ≤2y=30x>25 5 − 2x, 0 < x ≤2y= 50x>266−7x,0≤x≤7y= 60x>73 3 − 4x, 0 < x ≤4y= 30x>444−5x,0<x≤5y=40x>55 5 − 9x, 0 < x ≤9y=50x>977−3x,0<x≤3y=70x>34 4 − 9x, 0 < x ≤9y=40x>98 8 − 7x, 0 < x ≤7y=80x>777−5x,0<x≤5y=70x>54 4 − 3x, 0 < x ≤3y=40x>3ÌåäÒðÓàÌ ÂÌÈ-2ÝÀ1122244077−6x,0<x≤626 y = 70x>65 5 − 4x, 0 < x ≤428 y = 50x>45 5 − 3x, 0 < x ≤330 y = 50x>3ÌåäÒðÓàÌ ÂÌÈ-2ÝÀ55−8x,0≤x≤825 y = 50x>83 3 − 7x, 0 < x ≤727 y = 30x>77 7 − 8x, 0 ≤ x ≤829 y = 70x>8Çàäà÷à 7Ìåòîäîì Ôóðüå íàéòè ðåøåíèå óðàâíåíèÿ êîëåáàíèé ñòðóíû∂ 2U∂ U=äëèíû l = 2, çàêðåïëåííîé íà êîíöàõ: u(0, t) =∂t2∂x2u(2, t) = 0 è óäîâëåòâîðÿþùåé ñëåäóþùèì íà÷àëüíûì óñëîâèÿì:àU (x, 0) = f (x),Ê21(234f (x)x,0≤x≤12 − x, 1 ≤ x ≤ 20(−2x,0≤x≤12(x − 2), 1 ≤ x ≤ 20∂U (x, 0)= ϕ(x)∂tϕ(x)02x − x2 , 0 ≤ x ≤ 20x2 − 2x, 0 ≤ x ≤ 24167x,0≤x≤132−x, 1≤x≤230(−x, 0 ≤ x ≤ 1111213(2x,2(2 − x), 1 ≤ x ≤ 20x2− x, 0 ≤ x ≤ 2203x2 − 6x, 0 ≤ x ≤ 20154x − 2x2 , 0 ≤ x ≤ 216017x2 x− , 0≤x≤2420≤x≤1Ê1400à104x − 2x2 , 0 ≤ x ≤ 2x − 2, 1 ≤ x ≤ 2890ÌåäÒðÓàÌ ÂÌÈ-2ÝÀ5x2−+ x, 0 ≤ x ≤ 2208x − 4x2 , 0 ≤ x ≤ 20(−3x,0≤x≤13(x − 2), 1 ≤ x ≤ 20x,0≤x≤152−x, 1≤x≤2502x − ,0≤x≤13 2(x − 2) , 1 ≤ x ≤ 23042(18019x x2− , 0≤x≤224212223x,0≤x≤132−x, 1≤x≤230(242829x − 2, 1 ≤ x ≤ 20≤x≤1à(0x− ,0≤x≤12 (x − 2) , 1 ≤ x ≤ 220x,04x − 2x2 , 0 ≤ x ≤ 20x2 x− , 0≤x≤2422(2 − x), 1 ≤ x ≤ 226270≤x≤12x,Ê25−x,0(5(2 − x), 1 ≤ x ≤ 200≤x≤1ÌåäÒðÓàÌ ÂÌÈ-2ÝÀ205x,0≤x≤12 − x, 1 ≤ x ≤ 20x2 − 2x, 0 ≤ x ≤ 2002x − x2 , 0 ≤ x ≤ 2x2−+ x, 0 ≤ x ≤ 2204330(05x,0≤x≤15(2 − x), 1 ≤ x ≤ 2×àñòü II.

Òåîðèÿ ðÿäîâ1.1ÌåäÒðÓàÌ ÂÌÈ-2ÝÀŸ1 ×èñëîâûå ðÿäû×èñëîâîé ðÿä, ñõîäèìîñòü ÷èñëîâîãî ðÿäàÎäíèì èç âàæíåéøèõ èíñòðóìåíòîâ ìàòåìàòè÷åñêîãî àíàëèçà,óäîáíûì è ïîëåçíûì äëÿ ðåøåíèÿ ìíîãèõ çàäà÷, ÿâëÿþòñÿ "áåñêîíå÷íûå ñóììû" èëè ðÿäû.Ïóñòü çàäàíà ÷èñëîâàÿ ïîñëåäîâàòåëüíîñòü a1 , a2 , . . . , an , . . ..Ñîñòàâëåííîå èç ÷ëåíîâ ýòîé ïîñëåäîâàòåëüíîñòè âûðàæåíèåa1 + a2 + . . . + an + . .

. =∞Pann=1Êàíàçûâàåòñÿ ÷èñëîâûì ðÿäîì, à ÷ëåíû ïîñëåäîâàòåëüíîñòè íàçûâàþòñÿ ÷ëåíàìè ýòîãî ðÿäà.àññìîòðèì ñóììûS1 = a1 ,S2 = a1 + a2 ,...Sn = a1 + a2 + . . . + an ,êîòîðûå íàçûâàþòñÿ ÷àñòè÷íûìè ñóììàìè ðÿäà. ×àñòè÷íûå ñóììû îáðàçóþò íîâóþ ÷èñëîâóþ ïîñëåäîâàòåëüíîñòü S1 ,S2 , . . . , Sn , . . .×èñëîâîé ðÿä íàçûâàåòñÿ ñõîäÿùèìñÿ, åñëè ñóùåñòâóåò êîíå÷íûé ïðåäåë ïîñëåäîâàòåëüíîñòè åãî ÷àñòè÷íûõ ñóìì, ò.å.∃ lim Sn = S.n→∞44∞P×èñëî S íàçûâàåòñÿ ñóììîé ðÿäà. Äîïóñêàåòñÿ çàïèñüan = S , êîòîðàÿ ïðèäàåò ñèìâîëó áåñêîíå÷íîé ñóììû ÷èñëî-n=1ÌåäÒðÓàÌ ÂÌÈ-2ÝÀâîé ñìûñë.×èñëîâîé ðÿä íàçûâàåòñÿ ðàñõîäÿùèìñÿ, åñëè ïðåäåë ïîñëåäîâàòåëüíîñòè ÷àñòè÷íûõ ñóìì ðàâåí áåñêîíå÷íîñòè èëè íå ñóùåñòâóåò.àññìîòðèì ïðèìåðû.Ïðèìåð 1.111++ ...++...

=2·5 5·8(3n − 1)(3n + 2)∞P1.=(3n−1)(3n+2)n=1Ïðåäñòàâëÿÿ îáùèé ÷ëåí ðÿäà â âèäå ðàçíîñòèà111=−,(3n − 1)(3n + 2) 3(3n − 1) 3(3n + 2)âû÷èñëèì ÷àñòè÷íóþ ñóììó ñ íîìåðîì n 11 11 Sn =−+−+ ...+3·23·53·53·81111+−=−.3(3n − 1) 3(3n + 2)3 · 2 3(3n + 2)Ëåãêî ïîíÿòü, ÷òî ñóùåñòâóåò êîíå÷íûé lim Sn , ðàâíûéÊn→∞÷èò, äàííûé ðÿä ñõîäèòñÿ è åãî ñóììà ðàâíàÏðèìåð 2.1 + 3 + 5 + . . . + (2n − 1) + . .

. =Âû÷èñëèì ÷àñòè÷íóþ ñóììó ýòîãî ðÿäà.Sn = 1 + 3 + 5 + . . . + (2n − 1) =1.6∞Pn=11. Çíà6(2n − 1)1 − (2n − 1)n = n2 .245 ýòîì ïðèìåðå lim Sn = ∞, ñëåäîâàòåëüíî, äàííûé ðÿä ðàñõîn→∞äèòñÿ.Ïðèìåð 3.1 − 1 + 1 + . . . + (−1)n+1+... =∞P(−1)n+1.n=1∞Parctgn=1ÌåäÒðÓàÌ ÂÌÈ-2ÝÀÓ ýòîãî ðÿäà âñå ÷àñòè÷íûå ñóììû ñ íå÷åòíûìè íîìåðàìè ðàâíû 1, à ÷àñòè÷íûå ñóììû ñ ÷åòíûìè íîìåðàìè ðàâíû 0. Çíà÷èò,ïðåäåë ïîñëåäîâàòåëüíîñòè ÷àñòè÷íûõ ñóìì íå ñóùåñòâóåò, è ðÿäðàñõîäèòñÿ.Ïðèìåð 4. àññìîòðèì áîëåå ñëîæíûé ïðèìåð1.n2 + n + 1Äëÿ âû÷èñëåíèÿ ÷àñòè÷íûõ ñóìì âîñïîëüçóåìñÿ îðìóëîéa+b1 − abñïðàâåäëèâîé â ñëó÷àå, êîãäà a ∈ (0, 1) è b ∈ (0, 1).arctg a + arctg b = arctg1,31 1+37 = arctg 10 = arctg 2 .S2 = arctg1 12041− ·3 7nÏðåäïîëîæèì, ÷òî Sn = arctg, è äîêàæåì ñïðàâåäëèâîñòün+2àS1 = arctgÊýòîé îðìóëû, ïîëüçóÿñü ìåòîäîì ìàòåìàòè÷åñêîé èíäóêöèè.

ÏóñòüSn−1 = arctgn−1, òîãäàn+1n−11+ arctg 2=n+1n +n+1n−11+ 2= arctg n + 1 n + n + 1 =n−111−· 2n+1 n +n+1Sn = Sn−1 + an = arctg46= arctg= arctg= arctgÌåäÒðÓàÌ ÂÌÈ-2ÝÀ= arctg(n − 1)(n2 + n + 1) + (n + 1)=(n + 1)(n2 + n + 1) − (n − 1)(n3 − 1) + (n + 1)=n3 + 2n2 + n + 2n(n2 + 1)n3 + n= arctg=n2 (n + 2) + (n + 2)(n + 2)(n2 + 1)n,n+2÷òî è òðåáîâàëîñü äîêàçàòü.nÈòàê, Sn = arctg,lim Sn = limn→∞ππarctg 1 = . ÿä ñõîäèòñÿ, åãî ñóììà ðàâíà .44n+21.2n→∞n arctg=n+2åîìåòðè÷åñêàÿ ïðîãðåññèÿåîìåòðè÷åñêîé ïðîãðåññèåé íàçûâàåòñÿ ÷èñëîâàÿ ïîñëåäîâàòåëüíîñòüb, bq, bq 2 , . . . , bq n−1, .

. ., (b 6= 0, q 6= 0).Ñóììèðóÿ ÷ëåíû ãåîìåòðè÷åñêîé ïðîãðåññèè, ïîëó÷èì ðÿä∞Pn=12bq n−1 . Çàïèøåì ÷àñòè÷íóþ ñóììó ýòîãî ðÿäà Sn+1 = b + bq +àbq + . . . + bq n äâóìÿ ñïîñîáàìèÊSn+1 = b + q(b + bq + bq 2 + . . . + bq n−1) = b + qSn,Sn+1 = (b + qb + bq 2 + . . . + bq n−1) + bq n = Sn + bq n .Ïðèðàâíèâàÿ ýòè âûðàæåíèÿ b + qSn = Sn + bq n , ïîëó÷èì Sn (1 −q) = b(1 − q n ). Ïðåäïîëàãàÿ, ÷òî q 6= 1, âûðàçèì Snb(1 − q n )Sn =.1−q ñëó÷àå, êîãäà q = 1, î÷åâèäíî, ÷òî Sn = nq è lim Sn = ∞, ò.å.n→∞ðÿä ðàñõîäèòñÿ. Åñëè |q| < 1, òî lim q n = 0 è lim Sn =n→∞n→∞b, ò.å.1−q47b. Åñëè |q| > 1, òî lim q n = ∞n→∞1−qè lim Sn = ∞, ò.å.

ðÿä ðàñõîäèòñÿ. Íàêîíåö, åñëè q = −1, òî ÷àn→∞ñòè÷íûå ñóììû ïîïåðåìåííî ïðèíèìàþò çíà÷åíèÿ q è 0. Ïðåäåëðÿä ñõîäèòñÿ, è åãî ñóììà S =ïîñëåäîâàòåëüíîñòè ÷àñòè÷íûõ ñóìì íå ñóùåñòâóåò. ÿä ðàñõîäèòñÿ.Èòàê, ðÿä∞Pn=1bq n−1 ñõîäèòñÿ ïðè óñëîâèè |q| < 1, è åãî ñóììà1.3ÿäÌåäÒðÓàÌ ÂÌÈ-2ÝÀ∞Pbbq n−1 ðàñõîäèòñÿ.ðàâíà. Åñëè |q| ≥ 1, òî ðÿä1−qn=1àðìîíè÷åñêèé ðÿä∞ 1P1 111+ + + ...+ + ... =2 3nn=1 níàçûâàåòñÿ ãàðìîíè÷åñêèì. Êàæäûé ÷ëåí ãàðìîíè÷åñêîãî ðÿäà,íà÷èíàÿ ñî âòîðîãî, ÿâëÿåòñÿ ãàðìîíè÷åñêèì ñðåäíèì ñîñåäíèõ ñíèì ÷ëåíîâ11 11=+.an2 an−1 an+1Ïîêàæåì, ÷òî ãàðìîíè÷åñêèé ðÿä ÿâëÿåòñÿ ðàñõîäÿùèìñÿ.

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