Mathematic Solutions | University of California, Berkeley

University:
University of California, Berkeley
Course:
MATH 56 Linear Algebra
Academic year:

2021
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181

Pages:

4
Author:

larbiwxcvbn

1) a) Sn = 1/2^n. Prove (Sn) is a Cauchy seq, and hence a conv. seq. For m, n ∈ N, m > n, get only one "n"-index. |Sm - Sn| = |Sm - Sm-1 + Sm-1 - Sm-2 + ... + Sn-1 - Sn| < 1/2^m + 1/2^(m-1) + ... + 1/2^n 1 + r + r^2 + ... + r^n = (1 - r^(n+1))/(1-r) 1/2^n(1 + 1/2 + (1/2)^2 + ... + (1/2)^(m-n-1)) = 1/2^n * (1 - (1/2)^(m-n))/(1 - 1/2) = 1/2^(n-1) - 1/2^m |Sm - Sn| < 1/2^(n-1) - 1/2^m < 1/2^(n-1) Since |Sm - Sn| < 1/2^(n-1) ∀ m, n > N, and for ∀ε > 0, ∃N such that 1/2^(n-1) < ε, which means |Sm - Sn| < ε. b) No, counter-example: Sn = √n By definition of Cauchy, this example fails. 2) a) An = (-1)^n a) {1, -1, 1, -1, ...} is monotone b) {1, -1} is the subsequential limit c) Lim sup An = 1, lim inf An = -1 d) An does not conv. It oscillates b/w ±1. e) An is bounded. |An| = 1 b) Bn = 1/n a) {1, 1/2, 1/3, ...} is monotone decreasing. b) {0} is the subsequential limit c) lim sup Bn = 0, lim inf Bn = 0 d) Bn conv. to 0 e) Bn is bounded between 0 and 1. 3) Cn = n^2 a) {0, 1, 4, 9, ...} is monotone increasing. b) {∞} is the subsequential limit c) lim sup Cn = ∞, lim inf Cn = 0 d) n^2 does not conv. It diverges. e) n^2 is not bounded. 4) Dn = (6n + 4)/(7n - 3) a) Dn itself is decreasing, monotone. b) {6/7} is the subsequential limit c) lim sup Dn = 6/7, lim inf Dn = 6/7 d) (6n+4)/(7n-3) conv. to 6/7 e) Dn is bounded: [6/7, 4/3] 1) a) Sn = cos(nπ/3) when n = 6k, cos(6kπ/3) = cos(2kπ) = 1. Is monotone. b) {0, ±√3/2} is the subsequential limits. c) lim sup sn = √3/2, lim inf sn = -√3/2 d) Sn does not converge. e) Sn is bounded by [-1, 1] 2) tn = 3/(4n+1) a) tn itself is decreasing, so we can take itself as its own subsequence. Is monotone. b) {0} is the subsequential limit. c) lim sup tn = 0, lim inf tn = 0 d) tn converges to 0. e) tn is bounded [0, 3/4] 3) un = (-1/2)^n a) {-1/2, 1/4, -1/8, ...} is decreasing b) {0} is the subsequential limit. c) lim sup un = 0, lim inf un = 0 d) un converges to 0. e) un is bounded by [-1/2, 1/2] 4) vn = (-1)^n + 1/n a) when n = 2k, vn is decreasing → monotone b) {1, -1} is the subsequential limit. c) lim sup vn = 1, lim inf vn = -1 d) vn does not converge. e) vn is bounded by [-1, 1].