2. Consider the Fibonacci sequence {rn} given by x1 = 1, 22 = 1 and Xn = In-1 + In-2 for n > 3. Using Principle of Mathematical Induction show that for any n >1, *-=[(4725)* =(4,799"]
2. Exercise 2. Consider the sequence (xn)n≥1 defined by xn = Xn k=1 cos(k) k + n2 = cos(1) 1 + n2 + cos(2) 2 + n2 + · · · + cos(n) n + n2 . (a) Use the triangle inequality to prove that |xn| ≤ n 1 + n2 for all n ≥ 1. (b) Use (a) and the -definition of limit to show that limn→∞ xn = 0. Exercise 2. Consider the sequence (In)n> defined by cos(k)...
Given the logistic map Xn+1 = run(1 – Xn) with r > 0. Show the 2-cycle is stable for 3 <r <1+V6.
12 if x = 1,2 1. Define f:[0,2] → R by the rule f(x) = { 11 otherwise a. For any e > 0, find a partition Psuch that U (f, Pc) < € (be careful, as the minimum value for the function is one and not zero) b. Evaluate ſf
3. Using the Cauchy Schwartz inequality show that if u, veR" then ||u-<H4+ -
Define X1 = Z1, X2 = 22, ..., Xn = Zn and X = 36 L3fXi. Consider the following probability A=P('x=11<3). (d) Please provide the distribution of X and find the exact probability A (accurate to the third decimal place). (e) Please provide a lower bound for A by the Chebyshev's inequality.
(1) Suppose f :(M, d) + (N,0) is not uniformly continuous. Show that there exist an a > 0 and sequences (Xn) and (yn) in M such that d(Ion, yn) < and o(f(xn), f(n)) > € VnE N. (Hint: Negation of the definition of uniform continuity.)
define the sequence an as follows (3) Define the sequence an as follows Q1 = 1 and for n > lan = Van-1 + 2 (a) Compute the first four terms of the sequence (b) Prove an is increasing. That is, prove an < an+1 for all n € N. (c) Prove an < 4 for all n e N.
22.M. If c>0 and n is a natural number, there exists a unique positive number b such that b" = c.
The Ackermann function is usually defined as follows: In+1 A(m, n) = {Am - 1,1) ( Alm – 1, A(m, n - 1)) if m =0 if m >0 and n=0 if m >0 and n > 0. Use the definition of the Ackermann function to find Ack(3,2). Please show your work step by step.