Prove that each of the following sets is convex (a) {(x1, 22, x3) E R3 | 0 < 띠, x2, 23 and x1 + 2x2 + 3x3 6)
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
5. Partitions For each n e Z, let T={(x, y) + R n<I- g < n+1}. Is T = {T, n € Z} a partition of R?? Justify your answer using the definition.
1. Find the supremum and infimum of the following sets. (c) { (a) {, e} (b) (0,1) :n € N} (d) {r EQ : p2 <4} (e) [0, 1] nQ (f) {x2 : x € R} (8) N=1 (1 – 7,1+) (h) U-[2-7-1, 2”)
7. Find the interior and boundary of each of the sets(VR:nEN) and r EQ:0<< 2
please solve 2 to 6 with details Advanced Calculus: HW 3 (1) Suppose that a E R has the following property: for all n e N, a < Prove that a<0. (2) Prove that the set of integers Z is not dense in R (3) Let A = {xeQ: >0}. Determine whether A is dense in R, and justify your answer with a proof. (4) Find the supremum of the set A= {a e Q: <5} (5) Let a >...
No Contradiction 2. Let A and B be non-empty subsets of R, and suppose that ACB. Prove that if B is bounded below then inf B <inf A.
Find the interval of convergence. (Enter your answer using interval notation.) 27(x - 7)3n+6 n = 1 11 13 Use the equation 1 = Ï xn for 1x < 1 1 - X n = 0 to expand the function in a power series with center c = 0. 192 + 3x3 sW n = 0 Determine the interval of convergence. (Enter your answer using interval notation.) Use the formula In(1 + x) = - 1) - 1x = x...
. c) + < 2 b) 2 + 3x 27, 0. Solve for r: r' + 2.r < 2.1? +12
Please justify your answers : Which of the following sets are compact: i. {(z, y)E R 2 - 2y2 < 1} ii. {(, y) R22 < 2x2 + y2 < 4} ii. {(esin(ar), e cos(x)) R2: x 2 0}U {(x,0) E R2 : 0 iv. {(esin(0), e cos (0)) R2 x 2 0, 00 2r} x< 1}