P H A M N G O C T H A O (Chu bien) B A l T A P (Giao trinh Toan , nhom nganh I ) TAP HAI DAI HOC Q u c GIA HN TRUNGTAM thAngtin.THirVlfiN 510/33 V-GO NHA XUAT ban DAI HOC QUOC GIA HA NOI PHAM NGpC THAO (Chu bien) LE MAU HAI - NGUy I n VAN KHUE NGUy I n DINH s a n g - Bpl DAC TAC BAI t a p GIAI TfCN (Nhom ngdnK I) t a p HAI NHA XUAT b a n DAI HOC QUOC GIA HA NOI - 1998 L d l NOI DAU Giao trinh bM tap dudc chiing toi soan thao tUcfng ting vcti cac phan ly thuyet cua giao trinh giai tlch cua “Toan D cifdng" dung cho nhom nganh I Dai hoc Quoc gia H a Ngi N o la giao trinh chu yeu de sinh vien luyen tap sau hoc ly thuyet b phan I cac tap va cua “Toan D cUcfng” ve giai tich Trong no gom cac loai bai tap sau: ly thuyet gidi han, Topo va ham lien tuc R", phep tmh vi phan R" va iing dung, tich phan 16p, tlch phan suy rong va tich phan phu thupc tham so cung tich phan suy rong phu thuoc tham so'; chuoi so' day va chuoi ham; tlch phan bpi, tlch phan dvtdng va mat Trong moi chifdng, sau phan tom tat cac van de ly thuyet, cac cong thiic dupe dung c6 lien quan den chifdng do, chiing toi cho cac bai tap cP ban cua chucfng do, nghia la cac bai tap ma sinh vien can lam de n im dupe cac kien thiic va ky nang cd ban N g a y chung toi cung sip xep bM tap theo cac loai: ren luyen ky nang ap dung thuan tiiy cong thiic hoac ky nang tlnh toan, ren luyen tu suy luan va kha nang sang tao, nhutng bM tap mang tlnh chat ly thuyet, cac bM tap nang cao Mot phan khong nho bai tap moi chupng nham bo sung va md rong kien thilc cho sinh vien dieu kien thdi gian han hep ma viec day ly thuyet chiia thiic hien dUdc M ot so' bai tap kho dupe chiing toi danh dau (*) danh cho cac sinh vien kha gioi c6 y muo'n tim hieu sau them kien thiic va nang cao trinh dp cua minh Phan hu ng din va tra Idi cho cac bai tap cung diWc chung toi lam chi tiet va danh cong siic thoa dang Cac bai tap d l dUdc cho dap so' mot so bai tap kho dUdc giai ti mi M ac du vay sil han hep cua thdi gian va tru c nhu cau cap nhat cua sinh vien nen khong tranh khoi nhiing thieu sot qua trinh bien soan Chiing toi mong nhan dupe nhiing y kien dong gop cua cac dong nghiep va doc gia de hoan thien giao trinh hdn b cac Ian tai ban sau Nhan day chiing toi bay to su cam dn doi v i cac vien hoi dong tham djnh, dac biet la G S TS N guyen V an M a u cung cac P G S Nguyen Thiiy Thanh va Pham Chi Vinh da dong gop nhieu y kien cho sU hoan thien cua ban thao Hd ngi ngdy 18 thdng ndm 1998 C a c tac g ia G IA I H O A G H ir d N G D A N C h iid n g I G IC S l H A N A Gidl HAN CUA DAY a) Xet n+ 1 7n + L a y no = 5 < — < B, 7(7n + 2) 7n n >— 78 + 7c b), c) tUdng til a) 2.b) Xet X 2n n ^ < - > n -> 00 Tircfng til cho a) va c) a) Gia sik k la so' tu nhien nho nhat v i k > a K h i do: 0< a n n! a a n < a k-l a \ n-k+l ^ ( k - )! vk/ n -> o) b) Gia sii m la so' nguyen ducing v i m >k K h i do: in < — a" s m" n n a" m/ n -b " va m ,1 b = > 1, nhimg: n n 0< b" n [l+ (b -D n x1 < ->() n(n - l)(b - I) n->co c) Ttf; n n < suy ra: -> n -> CO n -1 d) Bkng qui nap ta c6 ; n! > U Ttf do: < < ^ / \n n n Ix / n ■> n ^ CO, \ 11 5.T if e - lim n -^ o o nen v i day {nk}, -1- - V n J -> + o o + e = lim k->ooV k -> + o o ta c6 : Neu { pk), Pk>l, Pk -> +°o k-> ++oc Bdi vi: 1+ nk + < 1+ + - V Pk^ Pk < 1+ V 1+ V — — nk + SUV : lim k->oc 1+ P, Tritdng hdp da}’ {qj,} tien oo 1+ -o o thi ap dung ket qua tren clio day: k ^ + 0 ta cung dUdc: Qk = e a) Ro rang x„ < Dong thdi: 1 ^ 1 X„ < + + — + h 1.2 (n - l)n 2 = n-1 n - 1n suj’ ra: x„ 1 Do ton tai Iftii x „ U^cc b) titdng W a) b) x„ = lo g (n + l) -> + 00 a) Day tang vi x^+i - x„ = n+l 10 10^ > va bi chan tren vi: 10" ■■■■=Po +i' l3 )“^ii±L X II +_ !_ > n+l l.V a y day t^ng TH: + ln ln x „ = In 1 1 , '^ 4; + + In _ 1+ 'V + = Vay x„ n^co Va x / = a + x„_, ditadenb® = a + b Do b^ - b - a = Tii: do: b - b )T a c 6; X2 = Ttfdo X2 - X , - - ( x o - X i ) ,X -X = - ( X i - X ) 1+ Vl + 4a Xi + Xn X + X] X3 = - ^ - — ^ Bang qui nap: k-2 Xk -X k -t = ( - ) X2 - X k-2 T\i b^ng phep cong lien ti§p ta c6; vdi k > ^ (-1) X,, “ X| - (X2 - X i) n-2 3.2 ^ i^n-2 ^2 '^'^2 -'^] 3.2 Do do: lim Xjj =, 2x2 11-2 n -> c c c) Ro rang x„ > vdi nioi n Bang qui nap c h t o g minh duac x„ < x„,, va x„ < v6i moi n Do ton tai lim x,, - a n -> o c x^_i Q ua gidi han dang tliiic x„ = - + — ta difdc a = — + — 2 2 ' Do a = d) Bang qui nap chtfng minh diWc daj- { x j giam va bi chan du i bcii do ton tai a = lim x „ , a > Qua gidi han d in g thiic n -> cc Xn = +x„_ ta CO a = a Tu’ a = > ^/5 e) Ta CO Xn = T\i x„ > x„,! va ton tai a = lim n —>-oc V X ,j_ i +Xj^_ t a CO v d i m o i n Qua gidi han d in g thxi^ a = -n/5 a) < Mq -t- q M q 11+1 1— q M In q b) sin(n + ) •^11+]) -'n I 211+1 , sin(n + p) < - + + ij+i n+p n+]) I 2*' 2n+p < < K n+1 Lay n > log;; c) < ^ii+p 1 (n + l)(n + ) (n + p)(n + p + ) n+ n+2 n+p n+p n+ n Lay n > — c 11 D^t Z„ - x - Xi l + l x - x l+ +lxn -X., J Khi v i moi n Zn Z„ Do ton tai lim Z„ Vay v i s > ton tai no cho n > no n —>oc p > ta c6 : 10 ^n+p )ds = j (P iy - Q.X5.)ds = dD ' dD ' j (Pdy - Qdx) = If cP 1) Vdx c“D + r?Q dyKJ dxdy dxdy = II divP dxdy D (Ch thich: V i vay cong thiic (G ) c6 t M xem la dang d so" chieii cua cony thiic (O -G )) 490 DS: (1) 3TtV2 -2 ( ) (3) (4 ) 491 Xet mot nguyen ham F(^,ri) bat ky cua f dol ’ isi bien n g h la la 248 -(?,r|) = f(=,iij oc va fff(q;Ti)d^ dr|= | - ^ d c d r i = |F(zri)d'n (p(D) q>(b) '' r^!M>(D)r Co the gia thiet det cp’(x,30 >0 V(x,y)eD Khi a[(p(D)] = cp(aD) va ta CO (Bai tap 470): j F i t , n) d^i = dD ciD) dx d-z = 0(nghia la rotffrac}t>= oV(j) e C ) Vky t|) F d r = t) grad(f) d r = c c 497 F n ds = ^ X y X.— + y.— + z.— ds a a ay = aj f ds = 4tc a^ 6D div F dxdydz D 5x dy dz — + — + — dxd yd z D '^dx dy dz) 251 ^ , divF =^(2xz) + dx cy dz‘ cy 5z = 2z + 1~ 2z = 1, dX) dxdy dz dxdydz = F n ds = D ( x ‘ +4y")0 (hinh ben) ta dxiOc 256 U(C,T1) = ' 271 71 neu ■neu (5.T1) e D (C.TI) g D neu (£.^) gL -»'7 MUC LUC TranI Ldi mS diu Ldi giai ho^c hu6ng din ■Chiitfng £ ■ ■ G lC il H A N ■A ^id i han cua day B Gidi han ham so, ham so lien tuc C h u a n g i l T O P O V A H A M L I E N T U C T R O N G R" C h u a n g J II P H E P T IN H V I P H A N T R O N G R" A H am kha vi tren R B H am kha vi tren R" C h iifin g r V W C H P H A N M O T l p - T IC H P H A N S U Y R O N G - T IC H P H A N P H U T H U O C T H A M S O - T IC H P H A N S U Y R O N G P H U T H U O C T H A M SO A Tich phan mot 16p B Tich phan suy rong 13 C Tich phan phu thupc tham so’ 15 D Tich phan suy rong phu thugo tham so" 15 Chuang V 258 C H U O I SO D A Y V A C H U O I H A M 16 A Chuoi so" 16 B D ay ham 18 C Chuoi h^m 18 ... 11 +1 1— q M In q b) sin(n + ) • ^11 +]) -'n I 21 1 +1 , sin(n + p) < - + + ij+i n+p n+]) I 2* ' 2n+p < < K n +1 Lay n > log;; c) < ^ii+p 1. .. n(ti+l) X „ = ( -1 ) 14 2" S in l^ ,y „ = ( -1 ) c o s '— (n = l ,2, ) 11 ( 11 + ) Khi do: x„ + y „ = ( - ) limx,, = - limy,, = - limx„ = 1, limy,) = lim(x„ + y „ ) = -l, lim(Xn + y n ) = 22 Cac phan... n < n -1 n n -1 n u-^oc B GI6 I HAN HAM SO, HAM SO LIEN TUC 36 B & X ^ uen c6 the' xem < x < K h i I 3x^ -2 -1 3x^ - 12 l = 3lx - 2l lx+2l < lx -2 l< s f e L a y = — ,1 V15 y 37 Coi -2 < X