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.
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2.
Exercise 2. Consider the sequence (xn)n≥1 defined by xn = Xn k=1
cos(k) k +...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all inte...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
1. Consider the sequence (an) with an = Vn2 + n - n, n = 1,2,3,,.......
1. Consider the sequence (an) with an = Vn2 + n - n, n = 1,2,3,,.... 1.1) Prove that (an) is an increasing sequence. 1.2) Prove that (an) has an upper bound, and therefore has a limit a 1.3) Find a, the limit of an when n + . 1.4) Using Definition 2.2.3 to prove lim an = a. n->00
Consider the sequence defined as a[1] = 2; and a[k] = a[k-1]+2*k-1 for all positive integer...
Consider the sequence defined as a[1] = 2; and a[k] = a[k-1]+2*k-1 for all positive integer k >= 2; . Show that a[n] = 1+sum(2*i-1, i = 1 .. n); . Hint: Start with sum(2*i-1, i = 1 .. n);and use the recursive definition of the sequence.
Use definition 89 to help me find the answer to Exercise 90 please. If you can...
Use definition 89 to help me find the answer to Exercise 90
please. If you can only do one of the problems for 90, do #2. This
would be much appreciated!
Definition 89. (Limit Superior and Limit Inferior) Let x be a bounded sequence and let (un) and (vn) be defined as above. Define the limit inferior and limit superior of x, denoted by lim infxn and lim supxn, respectively, as lim infxn = lim un N-00 and lim supxn...
1. Prove that if {xn} is a sequence that satisfies 2n² + 3 Xnl73 +5n2 +...
1. Prove that if {xn} is a sequence that satisfies 2n² + 3 Xnl73 +5n2 + 3 + 1 for all n e N, then {xn} is Cauchy. . Use the definition of limit for a sequence to show that 2. Suppose that {Xn} converges to 1 as n xn +1-e, as nº n
Due Friday April 12, 2019 in class 1. Consider a sequence an) defined by recurrence: a 1, and an ...
Due Friday April 12, 2019 in class 1. Consider a sequence an) defined by recurrence: a 1, and an a/(n-1) for n22. Prove using strong induction that an for any n2 1 2. Consider a sequence {an} defined by recurrence: a1 = 1, a2-1 and an-2an-1 +an-2 for n 2 3. Prove using strong induction that an K 3" for any n21
Due Friday April 12, 2019 in class 1. Consider a sequence an) defined by recurrence: a 1, and...
4. (20 pts) Let {xn} be a Cauchy sequence. Show that a) (5 pts) {xn} is...
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...
Let the sequence X be defined recursively by x1 = 1 and Xn+1 = Xn +...
Let the sequence X be defined recursively by x1 = 1 and Xn+1 = Xn + (-1)-1 for n 2 1. Then X n is a decreasing sequence. an increasing sequence. a Cauchy sequence either increasing or decreasing. QUESTION 12 Check if the following statement is true or false: COS n The sequence is divergent. True False
Exercise 15: Let (cn) be a sequence of positive numbers. Prove: lim infºn+1 < lim infch/n....
Exercise 15: Let (cn) be a sequence of positive numbers. Prove: lim infºn+1 < lim infch/n. n700 Cnn +00 What is the corresponding inequality for the lim sup?
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result i...
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that 4 #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
1. Consider the sequence (an) with an = Vn2 + n - n, n = 1,2,3,,.... 1.1) Prove that (an) is an increasing sequence. 1.2) Prove that (an) has an upper bound, and therefore has a limit a 1.3) Find a, the limit of an when n + . 1.4) Using Definition 2.2.3 to prove lim an = a. n->00
Use definition 89 to help me find the answer to Exercise 90
please. If you can only do one of the problems for 90, do #2. This
would be much appreciated!
Definition 89. (Limit Superior and Limit Inferior) Let x be a bounded sequence and let (un) and (vn) be defined as above. Define the limit inferior and limit superior of x, denoted by lim infxn and lim supxn, respectively, as lim infxn = lim un N-00 and lim supxn...
1. Prove that if {xn} is a sequence that satisfies 2n² + 3 Xnl73 +5n2 + 3 + 1 for all n e N, then {xn} is Cauchy. . Use the definition of limit for a sequence to show that 2. Suppose that {Xn} converges to 1 as n xn +1-e, as nº n
Due Friday April 12, 2019 in class 1. Consider a sequence an) defined by recurrence: a 1, and an a/(n-1) for n22. Prove using strong induction that an for any n2 1 2. Consider a sequence {an} defined by recurrence: a1 = 1, a2-1 and an-2an-1 +an-2 for n 2 3. Prove using strong induction that an K 3" for any n21
Due Friday April 12, 2019 in class 1. Consider a sequence an) defined by recurrence: a 1, and...
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...
Let the sequence X be defined recursively by x1 = 1 and Xn+1 = Xn + (-1)-1 for n 2 1. Then X n is a decreasing sequence. an increasing sequence. a Cauchy sequence either increasing or decreasing. QUESTION 12 Check if the following statement is true or false: COS n The sequence is divergent. True False
Exercise 15: Let (cn) be a sequence of positive numbers. Prove: lim infºn+1 < lim infch/n. n700 Cnn +00 What is the corresponding inequality for the lim sup?
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that 4 #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in...
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