1 answer

3. Consider the sequence (x,) with x, =3 defined recursively by the ruleX 4-x Explore the...

Question:

3. Consider the sequence (x,) with x, =3 defined recursively by the ruleX 4-x Explore the sequence with your calculator: a. 1
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.

Answers

{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 = 03 we have x1=3 x2 = - = 1. x1 > X2 So, xo) Xnti is true for n = 1.</p><p>On. xn. >1 ton n=1. — (1) Xn+1 Liet, we assume that; xn xxSo, {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 =© 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-

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