Cauchy sequences 4. D&D 2.8.D: If fXnYn1 is Cauchy, then there is a subsequence z', such that Σk21 Iznk -2...
4 a) Find a recurrence relation for an, the number of sequences of 1's and 2's and 4's whose sum is n and with no 21 subsequence. b) Find a recurrence relation for an, the number of sequences of 1's and 2's and 4's whose sum is n and with no 44 subsequence. Answer is a) an = an-1+ an-4 + an-2 - an-3, b) an = an-1 + an-2 + an-5 + an-6, please explain how to get it,...
4 a) Find a recurrence relation for an, the number of sequences of 1's and 2's and 4's whose sum is n and with no 21 subsequence. b) Find a recurrence relation for an, the number of sequences of 1's and 2's and 4's whose sum is n and with no 44 subsequence. Answer is a) an = an-1+ an-4 + an-2 - an-3, b) an = an-1 + an-2 + an-5 + an-6, please explain how to get it,...
5. (15) Show that Log(z) satisfies the Cauchy-Riemann equation in D polar form of the Cauchy-Riemann equations may be helpful.) C\Re(z) S 0. (Hint: The 6:02 PM 6/25/2019
5. (15) Show that Log(z) satisfies the Cauchy-Riemann equation in D polar form of the Cauchy-Riemann equations may be helpful.) C\Re(z) S 0. (Hint: The 6:02 PM 6/25/2019
Questions 33 to 35 refer to the following Longest Common Subsequence problem. Given two sequences X-XI, X2,..., ...., X and Y y, y......... ya. Let C[ij]be the length of Longest Common Subsequence of x1, x2,..., Xi and y, y,..... Then Cij] can be recursively defined as following: CO if i=0 or j = 0 Cli,j][i-1.j-1]+1 ifi.j> 0 and x = y, max{C[i-1.7].[1.j-1); if i j>0 and x*y 0 The following is an incomplete table for sequence of AATGTT and AGCT....
8. Prove that if two rational sequences (a)1 and (n)1 are equivalent, then (a) (an) is Cauchy if and only if (bn) is Cauchy. (b) (an) is bounded if and only if (%) is bounded.
8. Prove that if two rational sequences (a)1 and (n)1 are equivalent, then (a) (an) is Cauchy if and only if (bn) is Cauchy. (b) (an) is bounded if and only if (%) is bounded.
THE BOLZANO-WEIERSTRASS THEOREM
2.4.1 Determine which of the following are Cauchy sequences. (a) an = (-1)" (b) An = (-1)"/n. (c) an=n/(n+1). (d) an = (cosn)/n.
4. Consider the sequence {z,.) such that z1-0, z2-1 and æn-telths,Yn (i) Show that (n) is convergent by showing that the subsequence of odd-indexed terms is monotonic increasing and subsequence of even-indexed terms is monotonic decreasing (ii) Find the limit of {%) (Hint: Consider x,-h-i)
4. Consider the sequence {z,.) such that z1-0, z2-1 and æn-telths,Yn (i) Show that (n) is convergent by showing that the subsequence of odd-indexed terms is monotonic increasing and subsequence of even-indexed terms is monotonic...
4. Solve the following Cauchy-Euler Equations. Answers 2. y1)2 where d--2 2, or equaivalently, ydd2( 1)2
4. Solve the following Cauchy-Euler Equations. Answers 2. y1)2 where d--2 2, or equaivalently, ydd2( 1)2
4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is bounded. Hint: See Lecture 4 notes b) (5 pts) {Jxn} is a Cauchy sequence. Hint: Use the following inequality ||x| - |y|| < |x - y|, for all x, y E R. _ subsequence of {xn} and xn c) (5 pts) If {xnk} is a See Lecture 4 notes. as k - oo, then xn OO as n»oo. Hint: > d) (5 pts) If...
Please answer all parts.
(2) (a) Give an example of sequences (sn) and (tn) such that lim sn ntoo 0, but the sequence (sntn) does not converge does not converge.) (b) Let (sn) and (tn) be sequences such that lim sn (Prove that it O and (tn пH00 is a bounded sequence. Show that (sntn) must converge to 0. 1 increasing subsequence of it (b) Find a decreasing subsequence of it (3) Consider the sequence an COS (а) Find an...