3. (i) State precisely what it means that the sequence (x(")) DEN converges to the limit x = R². [5 Marks] (ii) In the Euclidean space R², find the limits of the following sequences (if the limit exists): (a) limn→∞ (n²+1, 1-n); (b) lim∞ (exp(-1/n), exp(-n²)); (c) lim∞ (cos(2πn/3), sin(2πn/3)); 4. In the Euclidean space R2, give example of a sequence: (i) which is located on the quartic parabola x2 many different points, and converges to the limit (2, 16) = [15 Marks] x, contains infinitely [10 Marks] (ii) which includes each of the points (1,0), (1,3), (½³) infinitely - 2 2 many times. [10 Marks]
3. (i) State precisely what it means that the sequence (x(")) DEN converges to the limit x = R². [5 Marks] (ii) In the Euclidean space R², find the limits of the following sequences (if the limit exists): (a) limn→∞ (n²+1, 1-n); (b) lim∞ (exp(-1/n), exp(-n²)); (c) lim∞ (cos(2πn/3), sin(2πn/3)); 4. In the Euclidean space R2, give example of a sequence: (i) which is located on the quartic parabola x2 many different points, and converges to the limit (2, 16) = [15 Marks] x, contains infinitely [10 Marks] (ii) which includes each of the points (1,0), (1,3), (½³) infinitely - 2 2 many times. [10 Marks]
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![3. (i) State precisely what it means that the sequence (x(")) DEN converges
to the limit x = R².
[5 Marks]
(ii) In the Euclidean space R², find the limits of the following sequences
(if the limit exists):
(a) limn→∞ (n²+1, 1-n);
(b) lim∞ (exp(-1/n), exp(-n²));
(c) lim∞ (cos(2πn/3), sin(2πn/3));
4. In the Euclidean space R2, give example of a sequence:
(i) which is located on the quartic parabola x2
many different points, and converges to the limit (2, 16)
=
[15 Marks]
x, contains infinitely
[10 Marks]
(ii) which includes each of the points (1,0), (1,3), (½³) infinitely
-
2
2
many times.
[10 Marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39a0c022-e49a-477b-ab1a-da6fafebfbe8%2Fa52d53fc-937d-406c-8896-770ae5be6ae2%2Fi6ep4jf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. (i) State precisely what it means that the sequence (x(")) DEN converges
to the limit x = R².
[5 Marks]
(ii) In the Euclidean space R², find the limits of the following sequences
(if the limit exists):
(a) limn→∞ (n²+1, 1-n);
(b) lim∞ (exp(-1/n), exp(-n²));
(c) lim∞ (cos(2πn/3), sin(2πn/3));
4. In the Euclidean space R2, give example of a sequence:
(i) which is located on the quartic parabola x2
many different points, and converges to the limit (2, 16)
=
[15 Marks]
x, contains infinitely
[10 Marks]
(ii) which includes each of the points (1,0), (1,3), (½³) infinitely
-
2
2
many times.
[10 Marks]
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