# Dãy dạng u_{n+1}=f(u_n)

Topic nay danh de trao doi cac bai toan gioi han cua day so kieu $u_{n+1}=f(u_n)$ (o day $f$ la mot ham so) mot kieu day xuat hien thuong xuyen trong cac bai thi HSGQG cua Vietnam chung ta. Theo toi thi chung ta chi can mot chut kien thuc ve giai tich la co the lam tot bai toan kieu nay. Toi se post bai dau tien som nhat co the. 😀

## 13 thoughts on “Dãy dạng u_{n+1}=f(u_n)”

1. trungtuan says:

Cho $I \subseteq \mathbb{R}$ la mot khoang va $\{u_n\}_{n=1}^{\infty}$ la mot day so xac dinh boi $u_1\in I,u_{n+1}=f(u_n)(n=1,2,...)$, o day $f$ la mot ham so tu $I$ den $I$. De nghien cuu tinh hoi tu cua day so tren chung ta thuong dung ket qua:
Dinh li 1.
a)Neu $f$ dong bien tren $I$ thi day $\{u_n\}_{n=1}^{\infty}$ la mot day don dieu.
b)Neu $f$ nghich bien tren $I$ thi hai day $\{u_{2n-1}\}_{n=1}^{\infty}$ va $\{u_{2n}\}_{n=1}^{\infty}$ la hai day don dieu nguoc chieu(neu mot day don dieu tang thi day kia don dieu giam).

Chung minh.
a) Neu $u_1\leq u_2$ thi boi $f$ dong bien tren $I$ nen ta co $f(u_1)\leq f(u_2)$ hay $u_2\leq u_3$,… Do do day $\{u_n\}_{n=1}^{\infty}$ la day don dieu tang. Tuong tu neu $u_1\geq u_2$ thi day $\{u_n\}_{n=1}^{\infty}$ la day don dieu giam.
b)Boi vi $f(f(x))$ la ham so dong bien tren $I$ va $u_{n+2}=f(f(u_n))(n=1,2,...)$ nen hai day $\{u_{2n-1}\}_{n=1}^{\infty}$ va $\{u_{2n}\}_{n=1}^{\infty}$ la hai day don dieu, chung ta cung thay $u_1\leq u_3$ khi va chi khi $u_2\geq u_4$, suy ra hai day so do la hai day don dieu nguoc chieu.$\Box$

Thong thuong doi voi cac bai Toan Olympiad thuoc kieu nay , viec xac dinh day so da cho bi chan hay khong la viec kha don gian, cung vay viec xac dinh xem day $\{u_n\}_{n=1}^{\infty}$ ( doi voi a) ) hay cac day $\{u_{2n-1}\}_{n=1}^{\infty}$ va $\{u_{2n}\}_{n=1}^{\infty}$ (doi voi b) ) co hoi tu hay khong cung don gian boi vi chung ta da co ket qua quan trong sau day:

Dinh li 2. Neu mot day so don dieu tang va bi chan tren hoac don dieu giam va bi chan duoi thi no hoi tu.

Day la dinh li quen biet nen toi se khong trinh bay chung minh cua no o day, ban doc quan tam co the tim chung minh do trong cac giao trinh giai tich, chang han W. Rudin.

2. Phan Sy Quang says:

Mot phan quan trong khac do la su dung bo de quen biet sau :
day $\{u_n\}$ xac dinh boi $u_{n+1} = f(u_n)$
Neu $f(x)$ co dao ham voi moi diem thuoc $\mathbb{D}$ , $|f'(x)|\leq\alpha < 1$ va phuong trinh $f(x)=x$ co nghiem duy nhat la $b$ tren $\mathbb{D}$ thi day $\{u_n\}$ hoi tu va $limu_n=b$

3. Phan Sy Quang says:

Bai 1.
Cho $c$ la 1 so thuc duong . Day so $\{x_n\}$ xac dinh boi $x_{n+1}=\sqrt{c-\sqrt{c+x_n}}$ Tim $c$ sao cho voi moi $x_0\in (0,c)$ thi $\{x_n\}$ xac dinh va ton tai gioi han cua $x_n$

4. trungtuan says:

Sau khi doc cac post tren can than cac ban thay la:

1)Neu xay ra truong hop a) trong Dinh li 1 thi khong co van de gi, neu xay ra truong hop b) thi sao? Khi do chung ta dung ket qua sau:

Dinh li 2. Day $(u_n)$ hoi tu khi va chi khi cac day $(u_{2n-1}),(u_{2n})$ hoi tu va co cung gioi han.
Chung minh dinh li nay danh cho ban doc nhu mot bai tap.

2)Neu day so da cho hoi tu thi tim gioi han cua no bang cach nao? Dinh li sau se giup chung ta :

Dinh li 3. Neu ham so $f$ lien tuc tren $I$ va day so da cho hoi tu toi $l$ thi $l$ la nghiem cua phuong trinh $x=f(x)$.

3)Dinh li 1 la mot cach tiep can tot khi tinh don dieu cua $f$ duoc tim thay don gian, hoac viec xet dau cua $f'$ la don gian( thuong thi ham $f$ la co dao ham tren $I$), neu tinh don dieu cua $f$ khong the tim thay don gian thi chung ta lam the nao? Khi do chung ta dung dinh li sau:

Dinh li 4. Neu $f$ thoa man cac dieu kien :
a)$|f'(x)|\leq q<1\forall x\in I$.
b)Phuong trinh $f(x)=x$ co it nhat mot nghiem tren $I$.
Thi phuong trinh $f(x)=x$ co nghiem duy nhat tren $I$ va neu ki hieu nghiem do la $L$ thi $\lim_{n\to\infty} u_n=L$.
Chung minh. Goi $b$ la mot trong cac nghiem cua phuong trinh do. Boi dinh li Lagrang ta co $|u_{n+1}-b|=|f(u_n)-f(b))|=|f'(c_n)||u_n-b|\leq q |u_n-b|\forall n\geq 1$. Do do $\lim_{n\to\infty} u_n=b$, va dinh li duoc chung minh.$\Box$

to Quang: Cam on ban vi da nhac toi Dinh li 4.

5. Phan Sy Quang says:

Cai nay tu file cua thay Nam Dung .
Dinh ly trung binh Cesano :Neu day so $\{x_n\}$ co gioi han la $a$ thi cac day so trung binh $\frac{x_1+x_2+...+x_n}{n}$ cung co gioi han la $a$ .
Dinh ly co dang phat bieu tuong duong sau day .
Neu $\lim_{n\to \infty}(x_{n+1}-x_n)=a$ thi $\lim _{n\to \infty}\frac{x_n}{n}=a$ .
Su dung dinh ly nay ta co the giai quyet lop bai toan dang $x_{n+1}=x_n\pm (x_n)^{\alpha}$.De tim so $\beta$ sao cho $\frac{x_n}{n^\beta}$ ton tai gioi han theo dinh ly Cesano ta chi can tim $\gamma$sao cho $x_{n+1}^{\gamma}-x_n^{\gamma}$ co gioi han $a$ .Khi do $\lim_{n\to \infty}x_n^{\gamma}=a$ suy ra $\lim _{n\to \infty }\frac{x_n}{n^{\frac{1}{\gamma}}}=a^{\frac{1}{\gamma}}$ .Khi do ta chon $\beta=\frac{1}{\gamma}$

6. trungtuan says:

Uhm, post co mot loi, nhung khong quan trong lam. Quang cho vai vi du ve cai do xem! Cam on ban.

7. Phan Sy Quang says:

Cai nay co kha nhieu ung dung hay , anh Tuan a. De lay dan chung em se post 1 VD ( nho cai nay em da kill nhieu bai ma bon ban em bo tay roi do :D)

Bai 2( VNTST 1993 ).
Cho day so $\{u_n\}$ xac dinh boi $a_1=1,a_{n+1}=a_n+\frac{1}{\sqrt{a_n}}$.Hay tim tat ca cac so thuc $\beta$ sao cho day so $\{\frac{a_n^{\beta}}{n}\}$ co gioi han huu han khac 0.
( Co VD hinh nhu la bai thi hoc sinh gioi cua Nhat Ban nhung tam thoi em chua nho ra ,khi nao tim lai tren mathlinks post sau)
Quang.

8. Phan Sy Quang says:

Day la loi giai van dung tu tuong cua Dinh ly trung binh Cesaro.Em post ra cho moi nguoi tien tham khao.
Dau tien ta chung minh rang $a_n\to \infty$ khi $n\infty$.
That vay: Ta co $a_{n+1}^2=a_n^2+2\sqrt{a_n}+\frac{1}{a_n}>a_n^2+2$
Do do: $a_{n+1}^2>1+2n$ ,suy ra dpcm.
Tro lai bai toan ,xet hieu :
$a_{n+1}^{\frac{3}{2}}-a_n^{\frac{3}{2}}=\dfrac{(1+x)^{\frac{3}{2}}}{x}$
Trong do $x=a_n^{\frac{3}{2}}$.Vi $a_n\to \infty$ khi $n\to\infty$ nen $x\to 0$ .theo quy tac L’Hipotale ta co
$\lim_{x\to\infty}\dfrac{(1+x)^{\frac{3}{2}}}{x}=\dfrac{3}{2}$
Tu do suy ra
$\lim\dfrac{a_n^{\frac{3}{2}}}{n}=\dfrac{3}{2}$
Khi $\beta >\dfrac{3}{2}$ thi gioi han bang $\infty$ khi $\beta <\dfrac{3}{2}$ gioi han bang 0.Vay $\beta =\dfrac{3}{2}$ la gia tri duy nhat can tim .

Nhan tien day anh Tuan gioi thieu va quy tac L’Hipotale luon nhi:D
Quang.

9. vvmath says:

@Phan Sy Quang:

Hy vong ban se co mot ket qua mo rong cua day Lipzit cho truong hop

$u_{n+1}=f(u_n; u_{n-1})$

Have fun!

10. psquang says:

@vvmath : Thuc su em moi chi hoc gioi han day so gan day thoi.vvmath co the trinh bay mo rong cua minh khong a.Chac do se la 1 mang ly thuyet quan trong boi vi em cung da tiep can moi mot so bai toan dang nay,thuc su no rat kho.

11. trungtuan says:

Bai 3. (Vietnam MO 2005, Bang A)
Cho $(x_n)$ la day cac so thuc xac dinh boi $x_1=a,x_{n+1}=3x_n^3-7x_n^2+5x_n(n=1,2,\cdots)$. Tim tat ca $a$ sao cho day so do hoi tu, va tim gioi han cua day trong cac truong hop do.

12. nhat phuong says:

Thua thay,xin thay chi cho em mot so cuon sach ve day so,gioi han.Vi phan nay em moi buoc dau tiep xuc nen con la lam.Cam on thay nhieu.