. tf I 2k+ll
0<a+b=4aalt ru chon 230 ludn o=L
(**) e ,.-'!!f.-'),. < 0 , dring vi k > l.(k +t)(k + a) (k +t)(k + a)
I <x ( n*33 .Yn> l. Bai torink6t thfc. D n+3
Thi drt 3. Cho day sii (x,) xitc dinh bdi
- -'r-
^l ._
^, ^n+l - -nn
Z
Tim gicri hsn cia day si5 dd cho. Phfrn tich, Ta c6 x,*, = f,(x,) v6i
.f,(*) : 2* - 4' * | ld him so aong bii5n tron R..
nx
Ta s6 tim a,b sao cho x, ) an+b,YnZft, d6
limx, = +oo. D6 quy nap thdnh c6ng, ta cin phAi /,L t- 1
co 2(ak+b)- "''' 2a(k+l')+b.
k(ak + b)
DiAu niy tucrng ducrng voi ak + b - a, :f, it
ak' + bk
e a2k3 + (2ab - o')k' + (b' - ab - 4)k-1 > 0.
Ngoii ra ta cin th0m di6u kiQn dC bu6c co sd tlring, viQc chgn a,b sao cho <li6u kiQn tr€n dugc th6a m6n, d6ng thoi budc co s0 dring thi n=1,
T6m l4i
n=2 L,hikh6 kh6n n6n ta tim tli6u kign cl6 bu6cco sd dring L,hi n=3, tfc h ry>3a+b. Khi d6 co sd dring L,hi n=3, tfc h ry>3a+b. Khi d6
6
a =l,b - 2,ko =3 th6a m6n c6c y6u cAu.T6m lpi ta <15 chimg minh x, > n + 2,Y n 23. T6m lpi ta <15 chimg minh x, > n + 2,Y n 23.
VQy limx, = *oo. 0
Thi du 4. (I/MO-201t) Cho dAy s6 (x,) xac dlnh 2n
x, :l; xn = , ::^, Q,, *...* x, + xr),Yn,. 2. (rr - rJ
Vdi mdi n>1, dfit !,=x,,1 -xn. Ch*ng minhritng ddy sO (y,) cd gi6i hsn hiru hsn. ritng ddy sO (y,) cd gi6i hsn hiru hsn.
Phdn tich. V6i mqi n) l, ta c6
*,,,=(t*1*4*{la ua y.
\ ,? n n)
765
Ta c6 v, | -rr2 =3:v. = L> V,:V,=- > y, nen ta dU 2 ,,,rt lg r2
dodn y,*., > y,,Yn2l (*). That vQy <lidu ndy
. n2+3n+3 n2+n+ltuong dufi1g vor -1, + tt' x,*t > ---;- xn tuong dufi1g vor -1, + tt' x,*t > ---;- xn
e (n' + l)(n2 + 3n + 3)r(n' + n + l)(n+ 1)2, dtng.
Tatim a,b saochox, ( an+b,Vn>fr, d.5
, _ . (an + b)(nl + n + 1) <fiLakl(a + b),y n 2 ko.
tn' 3
n
D6 quy n4p thinh c6ng, ta cAn phii c6 k +l\k'z +1) .
:ff(ak+b)<a(k+l)+b.
k'
Thflt v4y, tli6u niy tucrng duong v6i
-(a + b)kz - (a + b)k - b >- 0.
Ta cAn di6u kiQn a + b <0; b <0. Ngodi ra ta cdn
th6m tli6u kign dC bu6c co sd dring, do budc co sd
kh6ng th6 dung ldti n =1 n6n ta tim di6u kien de bu6c co sd dring khi r=2, titc lA' 4<2a+b.
Ching han a = 5;b ='6 thi x, ( 5n - 6,Yn22. Bdi toSn tld ilugc gi6i quy5t. tr
Thi dU 5. Cho ddy sd (x^) xac dinh bdi r. =2'. _. --il+t f, =x,*!-,Vn>1.
X,
Chung minh riing ccic ddy ,6 y, =L;r,, = r,, - ,
n cd gicti hqn hfiu hgn vd tim gi6i hqn d6.
Phfrn tich. Ta c6 y, = 2;y,*, = f,(y,),Yn ) 1 v6i
f-(x):* - 1 - 1. ld hdm tl6ng biiSn tr€nr n. x@ +l)