3 (i)g(x+T)>g(x)>0,x∈[a,+∞) (ii) lim g(a=0,E lim f(a)=0 (ii) lim f(a+T)-f(r)=l x-+∞9(x+T)-9(x) f(r) x→+∞g(x) 10、设函数∫满足:(i)-∞<a≤f(x)≤b<+∞,(a≤x≤b);(i)|f(x)-f(y)≤kx-y,(0< k<1,x,y∈la,b)设r1∈la,b,并定义序列{xn}:rn+1=f(xn),n=1,2, 试证明: lim n=x存在,且x=f(x) 1、设函数∫满足:(i)-∞<a≤f(x)≤b<+∞,(a≤x≤b);(i)f(x)-f(y)≤|x-y,(x,y∈ ab)设x1∈la,b,并定义序列{xn}:xn+1=7(xn+f(xn),n=1,2 试证明: lim n=x存在,且x=f(x) 12.设a=0,定义an+1=1+si(n-1),n=0,1,2,…试求:lim∑ (Hint: Let bn an -1. Then, it is easy to prove that bn <0 and bn bn+1 for any n E N+) 13*.设函数∫连续不减,取定x0,由此定义rn=f(xn-1),n=1,2,…。若ro=a时,{xn} 收敛,试证明当min(a,f(a)≤xo≤mar(a,f(a)时,{xn}也收敛 14*、若ε0,ε1,ε2,…中的每一个均取自-1,0,1三数之一,证明 an =:EoV2+E1V2+.+Env2=2sin/o 24 对每个n成立 15*、定义数列xn=yne12n,n=n!n2en,n=1,2,…试证区间(xk,)包含了区间 (xk+1,3k+1) 1(1)数列{n=(+)a+单调减少的充分必要条件是2 (2)数列{n=(1+ x)1+n 单调减少的充分必要条件是0<x≤2 17、设∫(x)在[a上二阶可导,f(a)<0,f(b)>0,对一切x∈[ab],f'(x)≥0>0.,f"(x)≥0 (1)今1=f(n+1=xm-(x,n=1,2,3…证明{x}收敛于f在a,列上的零点 (2)令y (b-a)f(b) (b-yn)f(yn) f(b)-f)3n+1=--(m)=1,2,3…证明{n}也收敛于f在 a,b上的零点 (3)证明由(1)、(2)求∫的零点等价于求()=x-()且(x)=x-了()-/的不动点 (b-a)f(r f'(a) (Hint of(2): Let y=f(6)J(6)-f(a (a-b). Find out the point of intersection I1 of the straight line with T-axes. After that, find out the point of intersection r2 of the new straight line with z-axes 18、设∫是定义在[a,b上的实值函数,又设∫在o处可微分,其中a<xo<b,并设数列{an}3 (i) g(x + T ) > g(x) > 0, x ∈ [a, +∞); (ii) lim x→+∞ g(x)=0, & lim x→+∞ f(x) = 0; (iii) lim x→+∞ f(x + T ) − f(x) g(x + T ) − g(x) = l.  lim x→+∞ f(x) g(x) = l. 10  f 2+(i) −∞ < a ≤ f(x) ≤ b < +∞, (a ≤ x ≤ b); (ii) |f(x)−f(y)| ≤ k|x−y|, (0 < k < 1, x, y ∈ [a, b]). x1 ∈ [a, b] 612$ {xn}: xn+1 = f(xn), n = 1, 2, ··· . 3 lim n→+∞ xn = x & x = f(x). 11  f 2+(i) −∞ < a ≤ f(x) ≤ b < +∞, (a ≤ x ≤ b); (ii) |f(x)−f(y)|≤|x−y|, (x, y ∈ [a, b]). x1 ∈ [a, b] 612$ {xn}: xn+1 = 1 2 xn + f(xn)  , n = 1, 2, ··· . 3 lim n→+∞ xn = x & x = f(x). 12. a0 = 0, 1 an+1 = 1 + sin(an − 1), n = 0, 1, 2, ··· . 3 lim n→+∞
n k=0 ak n . (Hint: Let bn = an − 1. Then, it is easy to prove that bn < 0 and bn < bn+1 for any n ∈ N +.) 13∗.  f 74(Æ x0 581 xn = f(xn−1), n = 1, 2, ··· ( x0 = a 6 {xn} 793: min(a, f(a)) ≤ x0 ≤ max(a, f(a)) 6 {xn} %79 14∗ ( ε0, ε1, ε2, ··· 8;9<=Æ: −1, 0, 1 ;<9 an =: ε0 2 + ε1  2 + ··· + εn √ 2 = 2 sin  π 4
n k=0 ε0ε1 ··· εk 2k  =: bn >;< n ?@ 15∗ 1$ xn = yne − 1 12n , yn = n!n −n− 1 2 en, n = 1, 2, ··· . 3/3 (xk, yk) ABC/3 (xk+1, yk+1). 16∗  1 $  an = 1 + 1 n n (1 + x n)  +,(=DE!>?F! x ≥ 1 2 .  2 $  an = 1 + x n 1+n +,(=DE!>?F! 0 < x ≤ 2. 17 f(x)  [a, b] 0GHI f(a) < 0, f(b) > 0, >9@ x ∈ [a, b], f (x) ≥ δ > 0, f(x) ≥ 0. (1) . x1 = b − f(b) f (b) , xn+1 = xn − f(xn) f (xn) , n = 1, 2, 3, ··· .  {xn} 79# f  [a, b] 0%J ξ. (2) . y1 = b − (b − a)f(b) f(b) − f(a) , yn+1 = yn − (b − yn)f(yn) f(b) − f(yn) , n = 1, 2, 3, ··· .  {yn} %79# f  [a, b] 0%J ξ. (3) 5 (1) (2)  f %JK# g(x) = x− f(x) f (x) & h(x) = x− (b − x)f(x) f(b) − f(x) LJ (Hint of (2): Let y = f(b) + f(b) − f(a) b − a (x− b). Find out the point of intersection x1 of the straight line with x−axes. After that, find out the point of intersection x2 of the new straight line with x−axes by using x1 replecing a, and repeat continuously the program. What is the signification of geometry?) 18 f !1 [a, b] 0ABC f  x0 MIDEE8 a<x0 < b 6 $ {an},