Question

3. Consider the sequence (x,) with x, =3 defined recursively by the ruleX 4-x Explore the sequence with your calculator: a. 1 STO X STO 3-X ENTER, ENTER ENTER ENTER. 4-x Apparently the sequence diverges / converges to b. State the MONOTONE CONVERGENCE THEOREM: c. Use induction to show that (x) is decreasing for all n when x, 3 d. Use induction to show that (x.) is bounded below by 0 when x,- 3. e. Conclude from (b-d): d. To find the limiting value L, realize that lim x, lim x=L Then use the recursion to find the value of L.

Answer 1

{en} be a sequence x1 = 3 X.7+1 = 4-xn. X = 3 x2 = 1. = 0.3333 4-1 1. 24 = = 0.272727 4-0 3333 X5 = = 0.268292 4~0.272727 = 0.26797385 4-0.268292 27 = 0.26795096 Aparantely the Sequence.converges to 0.267 6 Morrotone Convergence theonema A mortone ine neasing Sequeme e it bounded above then the sequenee is convergent and converges to it's least upper bound and a monotone decneasing Sequence it bounded below then it is convergent and Converges to the greatest Lower bound. Actually the theonem, says it a sequence is monotone and bounded them it converges

3 we have x1=3 x2 = - = 1. x1 > X2 So, xo) Xnti is true for n = 1. on. xn. >1 ton n=1. — (1) Xn+1 Liet, we assume that; xn xxn+is true for n=k - (2) We have to prove that, XK+1) XK+2 Now, K+1 Xk+2 1 4-XK+ 4- xk 1 4-Xk+ 4-XK. From (2) we have; XK XK+1 on. - XK - XK+1 on 4-ak < 4-XK+1 on. 4-XK. <1: 4-XK+I on. 4-XK+) 71 4-XK on xK+I 71 xK+2 on. Xk+1 > XK+2 Hemee by induction we see that XK+1) XK+2 that is ai>X27X37... > XK XK+1) XK+2

So, {Xn] is deen easing for all n cohern x1 =3 Using induction we have to show that {Xn} is bounded below by o when x = 3 x =3 For, n = 1; So; the statement 70. Liet, the statement holds for n=k ako. that is Now, XK+1 = _1. _ 4-Xk Xk> o on – Uk Lo OP, 4-ak <4. OP - K 7 1/4 on, XK+1 7+ - 0.25 on, xk+iyo The statement holds for n=k+1 So; {Xn} is bounded below by o cohen x, = 3 ② from 6 and ③ we have the sequence {xa} is montone decreasing and bounded below So, we can say that the sequence is convergent from Monotone Convergence theonem

© Liet, the limiting value bel then, lim an = lim until because the tail terms of Se queree Coill tunds to I a Now, en+1 = 4-Xn . no 3n+1 = 1 on lim xn+ 1 = lim an on 1 - ze on, l (4-1) = 1 o 46-l² = 1 xp. on, 1² 4 1+1=0 1 - 4 J44)24.1.1 4 7 √ 16-4 2 - 44 253 = 2113 I = 2+13 = 3.73 on l = 2-3 = 0.267 • we have 3 = x1)x27x37... >xx> XK+1 Sinee, all the terms of sequence less than 3 So; 17 3.73 so, laoill be 0.267 Con 2-3) The limiting value is 0.267 de