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Let (xn) be a sequence of strictly positive terms and let (yn) be the se- quence...
Problem 2 Let {xn) , (yn) and {zn^ be three number sequences and suppose that for some fixed integer K, we have Please check whether following statements are true or false. If true, prove them. If false, provide at least one counter-example. (i) The convergence of rn implies the convergence of yn (ii) The convergence of zn implies the convergence of yn. (ii) The convergence of n, zn to some number L implies the convergence of yn (iv) The divergence...
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
41. Let (an) be a sequence of strictly positive real numbers and Sn = ak (a) Suppose that the series Σ an/ S,. an is convergent, determine the nature of the series an is divergent, show that 00 (b) Suppose that the series 1 1 Sn-1 Sp an a/S Then deduce the nature of the series
41. Let (an) be a sequence of strictly positive real numbers and Sn = ak (a) Suppose that the series Σ an/ S,. an...
3. Give an example of a sequence {sn} that is not monotone, but the se- quence {s} is monotone. (7 points) carlo ST 4. Let $i = 4 and 9n+1 = (38m + 1)/5 for n 2 1. Show that the sequence {sn} is bounded and monotone, and find its limit s. (10 points)
Let an, hen be a strictly monotonic sequence of real numbers with aro and the limit of 1. Let Yn be a sequence of Continuous real-valued fmetions on (0,13 with Yn 20 and Jual, so that the support of Yn is in Jan, anta [ A function f : [0,13? → R is defined as: nein | 6%9) - 2(--) -.-) . . a) Cubabe si Cinsel)aly and . (5 testube. Tip: Fix xe [an, Amen] (then overy and do...
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
#s 2, 3, 6
2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...
- a) Let Xn be a sequence such that Xn+1 – xn| son for all n E N. Show that the sequence is Cauchy (and hence convergent). b) Is the result in part a) true if we assume that In+1 – 2n| <
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 Xi, X2, , xn be independent Normal(μ, σ*) random variables. Let Yn = n Ση1Xi denote a sequence of random variables (a) Find E(%) and Var(%) for all n in terms of μ and σ2. (b) Find the PDF for Yn for all n c) Find the MGF for Y for all n