Tài liệu bồi dưỡng học sinh giỏi chuyên môn toán

2,9 Giai Xet/„ = j H a i day tiep c a n n h a u (sinx)"dx,tac6: 9.1 Mot s6 kien thufc thifdng dung plnh n g h i a Hai day so ( x „ ) vd (2/„) gQi la tiep c^n niu /„ = - y Jcosxd(sinx)"~\ ( s i n x ) " - ^ d(cosx) = - ( s i n x ) " " * cosx|| + 0 /„ = ( n - l ) y " c o s x ( s i n x ) " - d x = ( n - l ) J /„ = ( n - {1 - sin^ x) ( s i n x ) " - ^ dx, D i n h l y 18 Hai day tiep cq,n thi hoi tu den cung mpt gidi han f f 1) y (sinx)"-=^dx - ( n - 1) J(smx)''dx 1)4 = (n - l)/„_2 - (n - Cbufng m i n h Gia sii ( x „ ) , (j/„) la hai day so t i l p ( x „ ) la day tang, (yn) la day so giam Ta cut cho i&ng ba dieu kien (1), (2), (3) dung tif n = (neu khong t h i t a load di mot so so hang) Xet day so (zn) dinh bcti = x „ - j / „ , V n = 1,2, V?Ly n/„ = (n - l ) / „ _ /„ = ^ ^ / „ _ V i /Q = | , / i = nen: 2n - 2n - 2n - , 2n • 2- ; r1- 22n n 23- " - 43- 21- TT 2' 2TI ' n - 2n 2n-2 2^ "'2n ^ _2n_ 2n-2 2n + l ' n - l " ' ' 2n + r n -l 2n + l Ta CO Zn+i - 2n = (x„+i - J/n+i) " (x„ - y„) = (x„+i - x „ ) + (y„ - j/„+i),Vn e N * Ma x„+i - x„ > 0, Vn = 1,2, va 2/„ - y„+i > 0, Vn = 1,2, nen Zn+i Do (16: ' 2n /2n-l ~ 2n+l' ViO+oo V2Hn ^ < l- l.TCr(*)suyra n-^+oo 2n + JSfoo (x„) tang va bi chan tren (bi c h ^ tren bcii yi chang h?in) nen hpi t u (yn) giam vk h\n dudi (bj ch?in diTdi bdi x i chang h^n) nen hpi t y n/2 =*-/2n+l < hn n—+oo x i < X2 < X3 < • • • < x„ < y„ < j/„_i < • • • < y2 < y i % 0< n—•-j-oo l i m „ , VA; = 1,2, hay ^fc 0, 6„ > (Vn = 1,2, ) T i t {al + bl) > (a„ + b„f \un-Vn\= n—>+oo lim ^ = n — ' + 0 n! • i 'I Po (Ufi) v a (t;n) l a hai day ticp can Suy r a hai day h o i t u den cung mPt so, t a k i hieu so l a e Ta cd suy r a a„ > 6„, Vn = 1,2, T a c6 " n < e < t^n, Vn = 1,2, On+i = Ta gia suf phaii chiing rang e la so hOXi t i , e = — , vdi p v a no la hai so t u -T- < a„ + 0„ — r — = a„, Vn = 1,2, a„ + &„ nhien, no khac khdng Ta cd Suy r a day (a„) la day so giam T a c6 1 p , 1 l + 7r + ;77 + - + — r < — < l + 77 + r T 1! 2! no! no l ! 2! + - + — + no! — : no! Nhan t a t ca cho no! t a dudc Suy r a day (6„) la day so tang Vay 6i < 62 < > l < p ( n o - l ) ! < A + 1, •• < 6„ < a„ < ••• < a i Do day so (a„) giain va bi chan dudi (bdi so 61), day so (&„) tang va bi chan tren (bdi so o i ) Vay ca hai day so da cho cung hpi t u D a t va lim a„=a l i m b„ = b, k h i tijf n-»+oo (*) A la so t u nhien, >l = no! + ^ + ^ H 1M a (*) khong the 1! 2! no! xay dUdc (vi A\k{A-\-1) la hai so t u nhien lien tiep, p(no - 1)! cung la so t u nhien) Dieu vo If cluing to e khong the la so hiiu t i V$,y e la so vd t i Bai toan Cho day so (a;„) nhii sau : 6„4.i = ^4^ (Vn = l , , ) cho n —• +00 t a dildc b = —^—, hay a = b Vay t a cd dieu phai chiing minh B a i toan 235 (De nghi t h i O L Y M P I C 30/04/2003) va (vn) xdc dinh bdi: H i f d n g dan Xet day so (?/„) n h u sau 2/„ = x„ + J ^ ' ^ ' ' ^ - = 1.2, • • K h i dd Wn = + Tf + ;7f + •• • + ^ , 1! 2! n! i;„ = u„ + - ^ , V n = 1,2, TV Chiing minh (u„) va (u„) cd gidi hQ,n chung la mqt so v6 ti G i a i u„+i -Un = -.—^—rr- (n + 1)! Chiing minh r&ng day so cd gidi han la mQt so vd ti Cho hai day so (u„) > 0, Vn = 1,2, Suy r a day (ti„) la day tang- ^^cng t u n h u bai toan 235 t a chiing m i n h dir0c ( x „ ) va (y„) la hai day tiep •^ta va hai day so cd gidi h^n chung la so vd t i ^ a i toan 237 Cho day so {x„}+f°i nhu sau: x„ = V ^ + - ^ , V n= l,2, ^ k\! k=o Ch ' ''^^9 minh r&ng day so cd gidi hfin la mgt so vd ti 342 343 HUdng d i n Xet day so " h u sau: {yn}n=i " V^y day (xn) tang va bi chan tren (bdi yi chang h^n) nen hOi tu, day (y„) giain va bi chan dudi (bdi x i chang han) nen hgi ty Ta c6 lim fc=0 y„ = lim n—+00 Khi titcmg tit nhit bai toan 235 ta chi'mg minh diroc (a;„) va {y„) la hai day tiep c§n va hai day so c6 gidi hgn chung la so v6 t i {yn}t^i Bai toan 238 Xet hai day so {xn}n=i 1+ - n—+00 \ = ny lim ( l + - ) n—+00 \ 71/ lim ( l + n-.+oo \ i)= Um x„ n-.+oo nj V^y ta CO difiu phai chi'mg minh nhu sau: isjhan xet 23 Tic (*) ta thay x„=(l + i)", iM=(l + i ) " ' (V„=.,,2, ) n+l ,Vn=l,2, ChUng minh r&ng: a) Xn < 2/n,Vn = , , b) Day (x„) tdng thuc si/ vd ddy (2/„) gidm thtfc stf c) Hai ddy so {x„}+~i, {yn}t=i cimg gidi han, duac goi Id so e Giai Bai toan 239 Cho hai ddy so (x„) vd (y„) nhu sau: a) Hien nhien b) Sur dung bat ding thiic Cauchy cho n + so, gom mpt so va n so + ta dildc: +n + - Mgitdi ta chiing minh dUcfc rang e Id so vd ti vde = 2,7182818284 Logarit c(j so e goi la logarit tu nhien vd ky hieu Inx = log^ x So e dong mot vai trd rat quan trQng todn hgc SvC dung {**) ta gidi dugc hai bdi todn kho {bai todn 239 vd bdi todn 240) < n + l a.,.a n + l x„ = l + i + + - l ^ - l n n , y„ = l + i + + _ l _ + i _ i n n ( n = l , , ) Chatng minh rdng hai ddy so da cho hoi tu 1+ - + n+l Hi/dng d i n Tfir (**) n h ^ xet 23 t a c6: dang thiic Cauchy cho (n + l ) so, gom mgt so va n so ta dUdc < l n e < (n + 1) In n\n^^^^ n Vay x„ < x„+i, Vn = 1,2, , nghia 1^ day (x„) tang thvfc sy L^i diing b a t Hay vdi mgi n = 1,2, ta c6: In ^ Vn = 1,2, n < va ^ < I n ^ Do vdi '^Wn = i , , t a c o ^n+i - x„ = in - ln(n + 1) + Inn = 2/n+i - y„ = \ J \ n - \ \ \ in - In n - ln(n + 1) + Inn = — n+\l > 0, In ^ ^ - ^ < '- n ta se chiing m i n h diMc hai day so (x„) va (y„) la h a i d a y tiep c a n n h a u Vay y„ < y„_i, Vn = 2,3, , nghla la day {y„) giam thi^c s\f c) Theo tren ta c6: Cling tien den ciing mot gidi han Tit ta c6 dieu p h a i chiing m i n h toan 240 Cho hai ddy s6 {x„} vd {y„} nhu sau: X I < X < • • • < x„-i < x„ < y„ < yn-i < • • • < y < yi, Vn = 1,2, 344 345 g l i i t o a n Xet day so { u „ } duac xdc dfnh bdi u„ = n^" (n g N*) DCit w„ = ^ 4^ = + • •• + , ^ =,Vn = , , ^" ^ n ( n + 1) v / { n + l ) ( n + 2) v^2n(2n+l) l i m x„ ud Hay tlm n—"+00 lim , y„ n—•+00 Chihig minh rdng day so G i a i TCr 16i giiii bai toan 239 t a c6: t,a gi(ii hO'^ '^^ ^ < l n ^ < i , V n = l , , + n Do do: - + n - + n n n { x „ } + f ^ c6 gidi han hQu h(in n tdng len vo h^n so vd ti G i a i Ta c6 n > , V n = 1,2, 1 , + •••+ — < In 2n n 1 , n + •••+ — > In + 2n n + Vay n - + l n , n + , , 2n , + In + • • • + In = n 2n-l , n + ,2n + l , + In + • • • + In — = n+ 2n 2n In -, n - n + l In l „ ? ! ! d l l < x „ < l n - ^ , V n = 2,3, n n - Hm (hi ^" lim ^ ^ =ln( ^ / HlLlLi^ \^n—'+00 n =ln2; / ta c6: sau: n X1+X2 = nen ciing hoi t u den mot so, suy day (x„) hpi t y x„,Vn > TiJ day sut dung nguyen l y kep suy n—'+00 thiCc vd ky hi^u - la day giam, >;r+T'^^^""^2n + l ' Do lim + - • •+x„+x„+i+- • • = ^ = J m ^ ( x i + X + - • •+x„) = 346 - - lim V —= lim — + • •+ ^Hm^l3^^' 347 ] + 2^ — Do WlU2 Wr-l , < "^"^w +i.Vj = 0,1,2, gdi vay nen vdi mpi j = , , , t a c6 " + +— ^ + y —• lim ' {U\U2••-Ur-l) Ihn V J - V l i m y^^^-^-'-^r-x Suy " hm > r i +^ ••• + ^_J_l>o — p < Suy U1U2 u lim y n—+00 ^-^ k=m }^hi A: nhan cac gia t r i , , , , n t a cho j = k, t a se c6: - i q l i m E — la mot so nguyen dudng, do J n (U1U2 • • - " r - l ) lim (2) n—•+(» U1U2 fc=m Hay (1) (2) suy ton t a i = r > di fc=m (urU2 Tiep theo t a se chilng m i n h V- 3m>2: "iô2 -Wm-l- lim 2^ - < fc=m d6 daii d i u mot diSu mau thuan Goi r la s6 nguyen dUdng nho n h i t cho r +j > + l , V j =0,1,2, Khidotaco i2' < r \ V i = l , , , r - l lim — < - Dieu mau thuan vdi (1) Vay a la so v6 t i L\iu y Sau day la bai toan 242, t6ng quat cua bai toan , day la bai toan rat sau sic va kho nen doi hoi ban doc phai c6 nang luc va sU kien t r i dang ke Tuy nhien c6 mgt thuan Idi la bai toan c6 nhieu l?lp luan da c6 bai toan trade Bai t o a n Cho {ukj^^i J»n = + 0 Chy:ng la day dan di^u tang minh rdng cdc so nguyen day so (a;„)+f°i, vdi V i vay U1U2 ••-Ur-l 2'+22+-+2'^ ^ " 2'-2 (n=l,2, ) fc=i Ur+j Ur+j x„ = vd gidi han cua day (a;„)+^i la rnqt s6 v6 ti '^•ai T a.,CO Suy x„+i - x „ = > , V n = 1,2, "n+l < 349 dMng m l a day s6 tang T a d n g t u nhit bai toan , t a cln'tng n i i n h cUtgc {x,y^i day c6 gidi h a n hCtu han = +00 Jiin nen J i m ^ = 0, f hVfc v$.y neu u i < 02 t h i clipn vi - Neu o i > a2 t h i J i m a*: = + 0 nen ^i fc du Idn t h i ak > a i , do chon r i la so nguyen duong nho nhat Idn hdn cho a i < a^, Thi^c ton t ^ i day v6 h^n cac so nguyen dUdng D o ton tai fci, fca, , fcn cho CO tinh chat nay, tiJc la O j < a^^, Vj = 1,2, , - De t i m {Tk)k=\ t a d p dyng each tren cho day so {an\X=n^ • Gpi r la so nguyen duOng nho nhat cho ar+j > + l , V j = : , l , , v a a j < a^, Vj = 1,2, , r - Dat a > lim = a, k l i i t f l x„ > 0,Vn = 1,2, va day (x„) tang suy TTIIP theo t a chimg minh a la mpt s6 v6 t i hQu t i : a = ^, vdi p,g N% Gia ngUOc 1^, a la Vi {w*}fc:^ 1^ = 1- Vdi m6i s6 tU nhien m > t a c6: ar = ( U r ) ^ < { U r + j ) ^ = ( « ' + / ) ^ Q lim — = I™ n-.+oo ^ Uk n->+oo fc=l : lim n—+00 — + ••• + yUl Urn , }_, +-J— lim > — n->+oo Ufc *:=Tn "»n-l + lim y "r+j / " lim y^-!- \ = V ^1^2••-^r-l lim k=0 lira *=0| (g + ) ^ + u,+jt KiU2 u„.-i.q^^lim^2->l Khi it nh?Ln cac gia t r i , , , , n t a cho j = fc, t a se c6: (1) fc=ni («.U2 , _0(^„hm,l.—y fc=m T i p theo t a chumg minh 3m > : u i u j " m - i - „hm^ 2^ fc=m = ^V^, = 0' ^' • • • " " dgn d i l u man t h u l n Dat {q + 1)2^+' la mot s6 nguyen duong, do ^ + iP^' Bdi v^y nen vdi uipi j = , , , t a c6 > ("iô2 a.-,) Ê (9 U1U2 - U r - l fc=m 71—•TJu k=m a2;y Do — E - = < Ur+j S u y N h u v^y u,u, - n - „ l i m V-2 r.2'-2 " lim = " r ^ ^ V j = , , , r - Viv?ly t a c o : V = '^'^'^ ''•'^^S faf^ •''o nguyen duong nen Ur < " r + j , do g' aj < a r , , V j = , , , n - - 350 y \- "•^^ (2) suy r a t o n t ^ m = r > d ^ ^ t i t gia t h i l t suy ^ lirn^ ak = + 0 se chi rang ton t?ii so nguyen dUdng r j cho lim / (uiU2 u„_i) " lim y"— \n-.^oo^^Uk "ay mau thuan vdi (1) Vay a la s6 v6 t i 351 \ +oo fe=rn+l r ni! lim y ( ^ Kl 3no e N* cho a„ = n - l , V n > UQ Gia s i i phan chiing r i n g vdi mpi no € N * , t o n t a i n > no cho a„ / n - D o a € Q nen a = —, vol = l ton t a i ni > 2M cho a „ , / n i - 1, tiic la < a„j < n i - Ta c6 - Vav < ( n i - 1)! l i m •/ -+ (ni-l)!—= - ni fc=ni f - ^ - An ! yl lim n-.+oo \^ni! K, M eN* Suy vdi moi n> M t h i n!a € N * Theo gia thiet phan chiing, k=l rii nj! + ni ni = < ni 5^ T T < l i d i i u mau t h u i n vdi n (n, - 1)! l i m T ^ e N\ fc=ni Ta CO ( n i - l ) ! a N* va vdi mpi n e ( , , , n i - 1} t h i ( n i - 1)! chia het cho n ! Do Man thuan nhan ditpc rhi'mg t o rKng 3no N* cho a„ = n - 1, Vn > no Dieu k i e n d u Gia sii 3no e N* cho a„ = n - 1, Vn > no K h i n (no + l ) ! a = (no + 1)! l i m fc=i V fc=ni ^ >i—+CX) ^ (ni - 1)! /J m-l = (no + 1)! l i m (ni-l)!a-(ni-l)!5;^^ kl n "0 , , fc! ^ 2^ fc! • no = M a t khac < o„i < n i - nen t a c6: lim T»-« + 00 k=ni ~ 1^ fc=ni+l fc=ni+l V Ti + (no + l ) ! hm = ( n i - 1)! ^ ^ M I m! + a„i + ni m! hm n-.+oo > , K! / n - + m hm n->+oo \ > n-+oo, fc=ni+l 352 _v^a.(no + l)! TT fe=ni+l —- fc=no+l n = ( n i - 1)! > T7 fc! fc! fc! / , , ni nm k=l no E / l i fc=no+l rn + ("0 + Kl ^ ^ > fc^ + 353 1)! h m —r - -r n-*+oo \no' n! — ^ ^ ajAno + ! ) ' • ^ Z-^ To + ) iV* C c & i t i m A cho T» fc! + Ad = A (a + Ac) j no- 2.10 so (i) x-„ X a c d i n h so h a n g t S n g q u a t c u a d a y so p h a n t u y e n t i n h C h u y 31 Neu bd = thi ta c6 the tuyen tinh hoa day so phdn tuyen tinh (xn) dd cho, rSi sau tim dU0c so hang tSng qudt cua day so (xn), lam cdch my nhanh gon hdn Ngodi ta cung c6 the tim dUdc so hang tong qudt cua day so phdn tuyen tinh bang nhftn.g cdch khdc hay han nila, ban doc co thi xem them bdi todn 30 d trang {phudng phdp ham lap) 2.10.3 D j n h ly 19 Cho a, 6, c, d € M cho ad - 6c ^ 0, c ^ Cho xi eR mqi n sau: Xet day so phan tuyen ti'nh { x „ } xac dinh bdi: M o t so t m h c h a t c i i a d a y so p h a n t u y i n t i n h 1,2, , ddt • va vdi , = x „ + i , neu no ton tai Xet ham so fix) nhU cx„ + d ax,i + ox + CX a, 6, c, d va p la cax hang s6 cho tnfdc Bang phudng phap quy na, ta c L g nunh duoc x „ ^ £ , V n = , , t o n g 2/1 = p , thoa man + d Chiing minh f Id song dnh I Cho day so ((„) duoc dinh nghia bdi: / yn+i = a y n + | ; n , V n = l , , zi = 1,| 2,^^, = c y „ + d2„,Vn = , , Vay van d^ lai la t i n i s6 hang tfing quat cua hai day s6 (y„):ri thoa man ( * ) T a t h u d n g dung haicax^hnha sau: e a c h (ap d u n g k h i (i) c6 h a i nghiem p h a n b i e t ) T a co y„+i + A2„+i = (a + A c ) y n + 354 (6 + Ad) 2„,Vn = , , tn+i = rHtn),yn (^n)»= = i,2, 'tdj/ CO the khong xdc dinh ki tit mgt thU tU ndo Con f'Htn) SQ,U: rVM = | x e R \ | ^ } : / ( x ) = 355 („} la tg.p V^y vdi f la song anh thi dang tMc 2) ditdc xac dinh neu va chi neu: XI (=*•) G i a Slit x„ (n Vfc = l , , , n - > 2) ditdc xac dinh K h i vdi mpi fc = 1,2, , Xk dudc xac dinh va Xk^h Xl= T a c6 xi 9^ ti Neu x i = t2 thi: r\h)=> f{xi)=^ti^X2 356 = ti n - cIo ta CO cac dinh If sau day ^ m h l y Neu day { x „ } hoi tu den L thi cL"^ + {d-a)L-b ^•^'J^g m i n h T I T X „ + I = " ' ^ " " ^ ^ , V n = , , cho n cx„ + d aL + b ^2 L = —— ^ cL' + {d-a)L-b cL + d ^ ' 357 = Q = + 0 ta dUdc D i n h l y Neu A = ( d - a)^ + 46c < thi day phan ki {khong hqi ty) D j n h l y 2 Gid sU A = {d - af + 46c > GQI a,P Id hai nghi$m phuang (ad - CUQ, {dn Id x) cx^ + {d - a)x - = Khi do: tfinh 6c)(xn - 0) ' {ad - bc){xn = a) £i = a vd chi x „ = Q , V n = , , 6) Old thiit X I 7^ a, dat Xn = p—^yn x„ - a Xn+i c) Gid thiet G N*, A = ^ cP + d Khi do: • Neu |A| = = AX„, Vn = , , n—'+oo l i m x „ = (3 _ n—•+00 Neu |A| = Xn = lim ca + d c.(i + d • Neu A = - vd xi AX„,Vn=l,2, n—.+00 < thi > vd x i — -J- TCr Xn = /3 7^ ky • Trudng hap A = A;/iong t h i ,i C h u f n g m i n h V i o, la iighi^ni ciia phirong t r i n h L = " f ' ^ nen cL + d , —„0 = ca + d lim Xn ^ ^ ^ ^ A „ - ^ lim ~ ^ = lim x „ = n — + 0 = ( d o Xi lim = = lim ~ ^ = /? n-.+oo Xn — 0) D o lim a - i - , _ Q- a0 + b l a i l a h i e n n h i e n T a d i m g p h u o n g p h a p q u y n a p G i a s i i x i = a K h i d o axi + aa + b cxi + d ca + d : = t a CO cp + d' a) T a c h i c a n c h i i n g m i n h n e u xi = a t h i x „ = a , V n = , , v i c h i l u ngUdc X2 = = 0 Q X,J:n V„.Q = x „ - , ^ x „ • yVeu A = - vd xi ^ f3 thi day { x „ } phan aa + b ~~ X„ - l i m X'^'^Xi n—+OC lim x„ = a P thi l i m x „ = (3 a = ~ a) c) T h e o k i t q u a c a n 6), ,suy r a A",, = A " - ' X i , V n = , , • N e u |A| < t h i l i m A " " ' = D o xi ^ a CQ + d cQ + d (•0 + d cn + d x „ - B J = • N e u A = - v a x i =^9 t h i x „ = / , V n G N * s u y r a • Neu A = - \h,xxi- 7^ v a thi X„+i a l i m x „ = ^fl ' n—'+00 i, =(-l)"Xi,Vn=l,2, Ta se c l u i n g m i n h d a y so ( y „ ) v d i y „ = ( - l ) " , V n = , , , k h o n g h o i t u (phan k y ) G i a siif x „ = a K h i d o axn + b an + b cx„ + a ca + d Cach T a c6 phan ky lim y i n - x n—>+oo = h m (-1) n—'+00 = - / = l i m 2/2n- V a y n—'+oo day Vay theo nguyen l y q u y n a p suy r a neu x i = a t h i x „ = a, V n = , , C a c h V d i m o i a thuQC K t a c l u i n g m i n h d a y (?/„) k h o n g h p i t u v e a N e u 6) T a " 7^ t h i |a - 1| = (5 > C h o n e = ^ |a - CO Xn+l = Xn+i - (aXn Xn+i - a +b \cxn + d a0 + b\ +b ''0 > n c h o y„u = 1, n g h i a l a aa + b c0 + dJ ' \cxn + d k h i Vn = 1,2, ton tai ca + d l2/no -a\ \\-a\>z ac0Xn + adxn + bc0 + bd- ac/3x„ - ad0 - 6cx„ - 6d {cxn + d){c0 + d) / acaxn V + adxn + baa + bd- a c a x „ - ada - 6cx„ - 6d^ icxn + d){ca + d) 358 , " o n e n d a y n a y k h o n g t h e h o i t u ve a N e u a- \i a 7^ - L y l u a n t i t o n g V u h u t r e n t a c u n g t h a y d a y (i/,,) k h o n g t h e h o i t u v e a V a y d a y ( y „ ) k h o n g (2/n) k h o n g h o i t y m a A : „ + I = 2/„Xi, V n = 1, , v a 359 7^ n e n d a y U d n g d i n X e m l a i b a i 2.G: "Sfr d u n g d i n h l i L a g r a n g e d ^ c h i h i g n u n h d a y s6 C O gidi h a n " d t r a n g 2G0 Neu / ( x i ) = ( ) thi a:„ = a : i , V n = , , • Khi f(xi) f(bi).f{x]) / Nni / ( n i ) / ( a ; i ) < tfii d&t a2 = oi, = xi, niu C h u y Gid siC phicang tiinh f{x) < fhl dnt (12 = X I , b2 = bi Dat birn ddi f{x) a2 + b2 1 Phuong Tiri) tuc qua tiinh tren, thu duOc day so • ChUny minh r&ng • ' tnnh J{x) = ado aia phUdng phdp goi Id phuong phdp lap ddn vd goi la ham Idp t o n ) Xet ham so f thoa man cue dieu kien: PhuOng trlnh f(x) —Ocd nhat; vi'fL a Id nghiein f'{x), f{x„J = ; Xn = x,^,Vn J^n+l = no + 1, no + , ('liifng minh rang , ^ t dA^• : ( a „ ) t n u g v a b i c u a (loan [a.,;6„j l a h ^ huTu h m — -,Vn J \Xn) N l i m x „ = n , vdi a Id nghiem cua phuong triiili J{x) — Cac t n t d n g licJp k l i a c dn'ruji, m i n h Ut()ug t i t T h c o bO d d t i c j n g IcJi g i i i i co d long • ' ' i t o a n 2 (J t r a n g 332, s u y r a t o n t a i c „ _ i n a m giilfa x „ v a x „ _ i nao c l i o b- a , "^^y 2" (>n-a„ = - J i a i K h o n g g i a m t n g q u a t , t a c o i / " niaui;; d a n d U d n g c o n / ' n i a n g d a u t r e n b S i 6, d a y so (6„) ^ a m va n d , t i bc^: « ; u y r a b a d a y s6 d o c = n—*-i-oc K h i (16 h i e n n h i e n l i m x , , = x „ „ , v i l a nghiem nao d o c u a phudug r t t n L l u , p (.,.) la d a v v o h , n D o t h ^ r K n g d a y so ( x J t a n g v a h / ( X „ ) ^ , V n = = , l , "liay x „ - = / ( X „ _ , ) Xn-i - - + / ' ( X „ _ , ) ( X „ - , ; „ _ , ) + ^ " ^ ' ^ ' - ' \ x n {!^^!'~^\o ( ) l a ditt.Jc /'U-.-i) /Un) - Xn^l)' = ^ ^ % ^ ( ^ - X ' „ _ ' - ' > ^ a \ („li „6„) = < / ( » ) = ^'71+1 1 ( P l n f d n g p h a j ) l a p d d n ) Gid sxt phuong khodng {a:b) x\ [a;6] ; trinh x = ;.) i d o / n g h i c l i b i e n n e n f{x„) CO Xet day so (x,,) nhxC sau: Xn+i — ly r a ( x „ ) l a d a y so t a n g G a ' V a y (k l a n g l i i O n i r i i a p h i w n g t i i n h f{x) - ngliirin (2) ,„„ „,„„,.;,,,„ = / U Bai loan (1) , l i m x „ = (^ K h i v i h a m / l i e n t u c v a / ( a „ ) / { t n ) < u e n n— + oc 1) diem Fourier) Xet day so ( x „ ) nhU sau: • N o u (.;•„) l a d a y d i r n g , t r o n g b a i t o a n nay, c6 u g h i a l a t o n t^i so t\f n h i e n Hi) c h o Dar, nghiem / " ( x ) Hen tuc r a gifc nguyen ddu tren doan [a;b] Chon XQ cho / ( x o ) / " ( x o ) > {diem X Q thildng ducfc goi la diem Giai • khodng ( a ; 6) Ta B a i t o a n 1 (PlutOng phap tiep t u y e u hay g o i l a plurong p h a p New- lim x„ = Q n->+oo = c6 nghiem - thdnh x — (l>{x) vd xdy dung day truy hoi nhu a bdi todn ^ nao cua phuong tn^ h u y Bdi todn 1 sau day duac xdy dung dua tren phucfng phdp yen hay goi Id phuong phdp Newton /(-'•) - 637 tiep B a i t o a n 114 Day ( x „ ) dxm cho nhu sau xy = ^"+1 2x3 S—, Vn = l , , 3x2-1' ~ 2x3 ~ " h,n + X > 0, k h i x e / V Vv/5'v/3 + a„ = a„ - a„_i < nen a„ ( t h i /(x) + - ^ ^ , suy > f{A) = -A'^ f{A) + A = O^A = lim \/5 n-.+oo a„ = " X < _1_ 75- Tit y'S' 1) Neu \a\ an vdi n nao t h i t a co -x^ + X - X ( x - ) x - ~ ~3x2 - - t h i f{x) /(a„+i) = - a „ suy K h i 23-3 •'"^'^^ ~ _Vi /{on) | x „ | = =^ l i m x „ == n-^-i-oo 4= -^^ '^^"^ v3 x e t t n t d u g h d p Xfc > Neu ^'^^^' Tritcic h e t t a c6 m o t so l u u y sau d a y : • X e t d a y so t r u y h o i t u y e n t i n h c a p l i a i Ui) = = thi xn = l,Vn^k, k+l,k + a, ui = Un+i 0; = u„_i aun - + 6, V n = l i n i ^ x,, = N e u (1) K h i 2, = a, V n Suy r a 1, , 1,2, ^ U ( ; +oo) t i n t i t b a n g t r e n s u y r a G Vay xfc+i = / ( x f c ) > V a i X > t i n < / ( x ) < X , s u y r a S u y r a d a y ( x „ ) h o i t u v a ^ ^ l i i n ^ ^ ' n = 1- V a y k l i i \xk\ ^ lim x „ = D a c b i r - t , n e u |a| > 4r tin v d i k n a o t l i i + +l - l i m x „ = •^(^,-, + - • X e t tntdng lidp < |a| < v a |a| ^ a „ , V t = , , D o QQ = -j= va d a y so ( « „ ) g i a i n n g h i e m n g a t ve n e n t o n t a i k h o a n g {ak;ak-\) chiifa lal •^ii'i+i = ul Un + \\in-\ '^"u+l "/.+ " M ("Wn+1 + nl + ul_i - "f, - W„ + h) U„ - aUnlln^ - 6u„ i ' n + 2M„ + U,^, - + = - au„+iu„ - = bu„ + i = K» til - (aUn - Un-l + b) U „ _ i - 6w„ + w„) h{un+i h( K „ Un+iUn-\ + Suy r a Taco f{\a\) e (-afc-2;-afc_i) |x2| = / ( ) = \x'i\ -/(l«l)e(afc-r,flfc-2) 3.4 -^o -/(l^^fcl)>ôo = T l i e o t r e n suy r a l i m x „ = ^ • n-^+oo SuT d u n g d a y so de x a y d L f t i g n g h i e m c u a m o t so phifdng t r i n h n g h i e m n g u y e n T r o n g b a i n a y t a q u y it6c n h i f sau: C a p ( x ; y) g o i l a n g h i f n i n g u y e n d u d n g c i i a p h u d u g t r i n h f{x, -/;((/„ + = a'^ + Z?'-^ - a o / i - ( r t + fl), Vn = , , U „ _ I ) la n g h i e m c i i a p h u d n g t r i n h x^ + x/ ~ axy - b{x + y) = a'"^ + /^'-^ -aa0-b{a k^+i| = l/U-fc)|- VS - « » , „ « „ _ D o d() v d i m o i so n g u y e n d i t d n g ii t i n {ak-2;(ik-i) \xk\ ( a i ; a o ) => / ( k ' f c l ) < / ( " i ) = V a y |xfc+i| > + y) = neu no thoa m a n phudng t r i n h va x, y la 640 (2) + 0) Q u a d a y t a l i i a y r i u i g n e u p h u d n g t r i n h c6 d a n g (2) t h i t a n e n x a y d i t n g d a y so C O d a n g ( ) B a i t o a n 1 Chi'fny vnnh nyuyc.n rang phmny trinh '— = cd vd han xy - • nqhiern J • dudny G i a i P h i l d u g t r i n h d a cho t u d n g d i t d n g v d i x^ + y^ - bxy = - ( d d a y a = 5, = O) (1) X o t d a y so (w„) n l u f s a u : U() = • ' , Ml = ; M „ + i = 5u„ - 641 Vn = 1,2, Un-i, „ / K h i = va ^^2±l+iirizi = 5, Vn = , , Do Un+1 + Un-1 _J}+1 'U'n+2 + U n IL_i = x^ + y^ +6 2 -n±£ B a i t o a n 117 Chiing minh rang phuang trinh sau c6 vd hg,n nghi$m ui+i - U„U„+2 = < - nguyen ^ xy U„_iW„+l G i a i Vdi x, y nguyen ditdng t h i phitdng t r i n h da cho tUdug dUdng vdi It^ - U„_lU„+i = - U„-2U„ = • ••= U i - UoU2 = -5 - 8xy = - Do u\ u„_i (5u„ - K „ _ i ) = - =!• + Xet day so (u„) xac dinh bdi _ 5u„u„_i = - , nghia la (u„; U n - i ) la nghi^ni ciia (1) vdi mpi n = , , V i day ( u n ) tang nghiem ngat nen tap hdp { u i , U , } la tap v6 lian cax; so nguyen dudng nen suy r a phildng trinh da cho c6 v6 han nghiem nguyen dUdng L i i u y Do {x\y) = (1;2) la mot nghiem ciia phUdng trinh da cho nen ta chon uo = 1, ui B a i t oa n 116 Cho so nguyen duang a > Chitng minh rdng phuang sau CO v6 han nghiem nguyen duang trinh Uo = ' ' 1, u i = 1, ,!&,:.' "n+l + U n - ^ Un+2 + U n _ ^ Un (Un+1 Un+l + U „ _ i ) Un+l - a va + Un-1 _ Un+2 Un Do a > nen day (u„) tang nghiem ngat, do tap hdp {{un-i;Un)\n e N * } c6 v6 han phan tiTf, sny r a di6u phai chiing minh hvtu y Bai toan 118 dUdc phat bieu dudi dang: Chiing m i n h rang vdi moi so nguyen dudng x vh y thi — luon la so chinh phudng xy + 642 nquven = 3xyz x'^ + y'^ + l = 3xy = ( u „ + + U „ ) Un U^+l = Un+l^n-l - Un-i) ul G i a i Chon = 1, t a cd Un+l + Un-l )Un+l - Uj = B a i toan 118 Chiing minh rdng phuang trinh sau c6 v6 han nghiem duang x"^ +y^ + Un+l ^l^n = Un+lUn-l Un = U U - Hy ix,y) = ( u „ _ i ; u „ ) la nghiem cua phUdng t r i n h da cho vdi moi n > t)ay ( u „ ) tang nghiem ngat, do tap hdp { ( u n - i ; Un)|n e N * } cd v6 l i ^ n ^han t i i , nen suy dieu phai chiing minh = a\) = a^Un - = (u„+2 + U „ ) U „ =^ (8u„ - u„_i) u„_i -ul = G i a i Vdi cac so nguyen diTdng x, y thi x'+y'~a'xy = Ta C O U = va =>Un+lU„_i - ^ = a\) xxj + Un+l Xet day so ( u „ ) xac dinh bdi Uo = l , u i = l , u „ + i = 3u„ - U n - i , V n > Khi U = va day (u„) tang nghiem ngat Ta cd W n + l U „ _ i ~ul = U2U0 - uf = (3ttn - U „ _ i ) U n - - U ^ = Do tap hdp { ( u n - i ; u„; l)|n € N * } cd vo han phan t i i , nen suy dieu phai cluing m i n h 643 B a i t o a n 119 duang Chiing minh r&ng phuang trinh sau c6 vo han nghifm x'^ + y"^ + z'^ = G i a i D?Lt x = 3xi,y xi + y'i + z^ +tl xyz 7io = 1, = l , u „ + i = 3u„ - u „ - i , V n > Theo bai toan 118 suy {xi;yi;zi) trinh = ( u „ _ i ; u „ ; 1) la nghiem cua phuong „ ^ xi +yi + zf ^ ixiyizi - • (1) ' = 1, Un+l "1 47t„ - 7i„_i, V n > = Theo bai toan 120, suy {xi;yi; zi\ti) = (u„_i; u„; 1; 1) la nghiem ciia phUdng t r i n h (1) Vay {x,y,z,t) = (27i„_i; 2(i„; 2; 2) la nghiem cua phuong trinh da cho vdi moi so t i t nhien n > Ta co dieu phai chiing m i n h i - Xet day so (u„) xac diuh bdi uo = l,ui ft = Axiyiziti Xet day so (u„) xac dinh bcli I = 3yi, z = 3zi Ta c6 vso &M i* c | G i a i Dat x = 2xi, y = 2j/i, z = 2zi, t = 2t\, ta c6 nguyen B a i t o a n 122 Cho so tu nhien n > Chiing minh r&ng phuang trinh CO vo han nghiem nguyen duang if + + \- x'n^ sau nXiX2 • • X n - Suy (x; y; z) = (3w„_i; 3u„; 3) la nghiem ciia phitdng t r i n h da cho vcli rnoi G i a i Chon X3 n > T i i day ta c6 dieu phai chiing minh B a i t o a n 120 duang = x„ = 1, ta c6 Chtcng minh fang phuang trinh sau c6 vo han nghiem nguyen x'f + xl + n - — nxiX2- Xet day so (u,„) xac dinh bdi x^ + 7/2 + 4-1^ = Axyzt Uo = l , i i = l , u , H + i = num G i a i Chpn z = < = 1, ta c6 x'^ + i/ + = Axy Wm+l"»u-l {nUm Uo = 1, "1 = l , W n + i = 4u„ - u„_i,Vn > K h i U2 = va day (w„) tang nghiem wghi Ta c6 Un + \ U n - l - ul = U2U0 - u\ ^ (4«„ - u,j_i)u„_i - wif, ^ Vay {x\y\z\t) = { u „ _ i ; u „ ; 1; 1) la nghiem n i a phUdng t r i n h da cho vdi moi 71 > Do tap liop { ( u „ - i ; i i „ ; l ; l ) | i ^ N ' } c6 vo han phan tiJt, nen suy i ' ' dieu phai cluing rninh - ?im-i,Vm > K h i 7^2 = n - va day (u,„) tang nghiem ngat Ta c6 Xet day so (w„) n h i i sau B a i t o a n 121 duang • - - "m = "2^0 7Xm-l) W m - l + "m-1 + n - = - uf„ u\ = n n - 2 nU,nUm-l Vay ( x i ; X2; xz\ •x„) = (u,„_i; 7/„,; ; ; 1) la nghiem cua phiidng t r i n h da |Cho vdi moi n > Do tap hop {{um-\\Um\; • • •; G N ' } c6 vo han phan tir, nen suy dieu phai cluing m i n h • B a i t o a n 123 duang Chiing minh r&ng phuang trinh sau co vo han nghiem x2+/ + 10 = ( x + l ) ( y + l ) G i a i PhUdng t r i n h da cho tUdng diidng Chiing minh rdng phuang trinh sau c6 vo hg.n nghi$m nguyen x^ + i/ + z^ + t^ = 644 xyzt x^ + 2/2 - 3xy - 3(x + y) + = Xet day so (u„) xac dinh bcli Uo = 1, u i = 1, u„+i = 3u„ - u„_i + 3, Vn > 645 nguyen K l i i c!6 U2 = va day (u„) tfiiig nghiem ngat Ta c6 Un+2 +Un ( U „ + + U„+2U„ - - _ U„_l - U n + + Un-1 3) Un+\ {Un+2 W^+i + 3u„+l = " _ Vay {x;y) = (u„-i;u„) la nghiem ciia phUdng t r i n h da cho vdi moi 7i > Day (u„) tang nghiem ngat, do tap hdp {(u„_i;u„)|n e N*} c6 v6 han phan t\t, suy difiu phai chi'mg minh ^ + Wn " 3) U „ U„+iW„-i - u\ B a i toan 125 Chiing minh rdng phuorig trinh sau c6 v6 han nghiem duang ' 3u„ nguyen (3u„+i - it„ + 3) u„ - tt^+i + 3u„+i = (3u„ - l i n - i + 3) u„_i -u\ 3u„ {Un+2 + Un + U^+i - U„+i = ui + ul_i - 5u„u„_i + (u„ + U n - l ) = u i + U Q - 5wo"i + (uo + wi) = =^«n + " L i - 5u„u„_i + (u„ + U „ _ i ) = = " ' K h i U2 = va day (u„) tang nghiem ngat Ta c6 - 2U„ - 2u„+i = (5u„ - u„_i - 2) u„_i - 1, Un+l ' + UQ - 1) Un-l - U^ - u „ U „ _ i + Un + Un Un-l 4uiUo + U l + Uo = nguyin G i a i Phildug t r i n h da clio tiidng dudng vdi 2) U „ 2Un+\ U „ + l U „ - l - (5u„+i - u„ - 2) u„ , • Vay (x;2/) = ( u „ _ i ; u , i ) la nghiem ciia phudng t r i n h da cho vdi mpi n > Day (u„) tang nghiem ngat, dp dp tap hdp {(u„_i;u„)|n e N * } cp vp h9,n phaii tijf, suy r a dieu phai chiing minh Xet day so (u„) xac dinh bdi uo = , u i = •, Xet day so (u„) xac djnh bdi UO = B a i t o a n 124 Chiing minh rhng phuong trinh sau c6 v6 han nghiem duang V - + 2/^ - 4x2/+ (x + ?/) = Suy r a +1/) = -25 {•': Xet day so (u„) xac dinh bdi Uo = 1, U l = \,Un+l = 7U„ - Un-l + 10, Vn > K h i d ' day (u„) tang nghiem ngat Ta c6 U n + 4- U n - 10 ^ Un+l + U n - l - U„+l U„ 647 10 ^ ^ (Wn+1 + Un-1 - tdng vdi 10u„ 7u„U„+l - 10(M„ + U „ + i ) - 7u„M„-i 10 (u„ + - 7u„U„_i - x^+y^-Axy + Ax-Ay , v + A= ^ (x - 2yf -Zxf + A{x-2y)+Ay j i ^ l ^.^jjjj, • , ' ' + A=Q ^ (x - 2y + 2f - 3y^ + Ay = , tt„_i) 10 ( u „ + • > , i ; i;,; 4- 10u„ - - - «„ + 10M„+I ' ^ n - l + 10) U „ - i - «>U^4.1 + =^ul 10) , , 2, phUdng trinh mot nghiem nguyen duong {x; y) thoa man cd x vd y dc.u le T h a t vay, v d i n = , t a c6 X = le t h o a m a n Xn — — — — , N h u vay, t a c d i e u p h a i c l u i n g i n i n h [x^ - llyn x „ - 2/„ ^ , s u y r a yn 7^ S u y r a Xn ! Xn-\-yn i c u n g l a so le S u y r a ^ ^ J } ^ = ^^^^^-^ + Sy^ c u n g l a s6 le H o n n i t a t a / 152/^)=4"+i — ^ - = Xn l a so le n e n m o t t r o n g h a i so le t i n , = 8y„ H le , N i u ^^i±yi^ ^ = Xn l a SO Ic u c u u i o t t r o n g h a i so N l u i ^ I L _ J ^ le t h i ^ ! L _ l I y ! i Vi , 172/^)1 = " + i 659 Xn - Vn , 1.3.6 X a c d i n h d a y s6 1.1 Xac dinh day so bang phUdng phap quy nap, phUdng phap d6i bien 1.1.1 1.1.2 2.1 Si'l dung dinh nghia gidi han day so 97 2.1.1 D i n h nghia gidi han day so 97 2.1.2 Gidi han rieng, day 98 2.1.3 Tieu c h u i n Cauchy 105 2.1.4 M p t so bai toan ap dyng 106 2.3 3 Xac dinh day so bang plnrong phap quy nap Xac dinh day so bang phUrtng phap d5i bien (dat an phu) Cac dong nhat thufc b6 sung 1.1.4 M o t so phep d6i bien dUdc djnh hudng bcli cac cong thiic luong gidc 1.1.5 1.1.G 1.2 1.3 10 'Phirong phap ham lap 2.2.1 PhUdng phap 2.2.2 Cac ket qua thirdng dung 2.4 39 M o t so utng dung ciia sai phan 42 1.2.1 M o t so dinli nghla 42 1.2.2 1.2.3 T i n h chat T i c h phan bat djnh 43 43 1.2.4 PhUrtng phap tich phan tifng phan 44 1.2.5 Phiidng phap he so bat dinh 45 1.2.6 T i n h tang 4G Xac dinh day so bang phildng phap sai phan 49 1.3.1 Sir dung phudng t r i n h sai phan tuyei t i n h cap mot 50 1.3.2 Siif dung pliitdng t r i n h sai phan tuyen t i n h cap hai G5 1.3.3 Sii dung phUdng t r i n h sai phan tuyen t i n h cap ba 1.3.4 Phifdng t r i n h sai phan tuyen tinh vdi he so bien thien 1.3.5 Tuyen t i n h hoa mot so phitdng t r i n h sai phan PhUdng phap xay di^ng day so nguyen ^4 2.5 2.6 115 nlnJ