(a) Show that each of the following sequences (an) converges to the given limit a. You may use any result from lectures, provided that you state it clearly. (i) (ii) n →0. n² - 2n+2 2n² (iii) π + sin(n) n 1 → →π. (b) Suppose that an → ∞ and bn → →∞. Show that anbn →→∞. (c) Prove that if (an) is Cauchy then (a²) is Cauchy. [

(a) Show that each of the following sequences (an) converges to the given limit a. You may use any result from lectures, provided that you state it clearly. (i) (ii) n →0. n² - 2n+2 2n² (iii) π + sin(n) n 1 → →π. (b) Suppose that an → ∞ and bn → →∞. Show that anbn →→∞. (c) Prove that if (an) is Cauchy then (a²) is Cauchy. [

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 72E

See similar textbooks

Similar questions

Determine if the following sequences converge or diverge. You must justify your answers. If they converge, find the limit. You may use any test/theorems/results from the lectures, but you must precisely state which ones you are using. an = 3n · In(sin(;)+1). 1. 2n2 +(-1)"-n 2. n2 (1+ 2)" + - 3 2n n 3. Cn = n 3n •
2. Determine whether the following sequence converges, and if s0, find its limit. (a) {Vn² + 3n – n} (b) {(1- ;)"} sin n (c) n
4. Let (n) be an increasing sequence of real numbers. Assume, furthermore, that (x₂) has a convergent subsequence (n). Prove that (xn) converges, and that lim (ân) = lim (Xnk).
Find the set A on which the sequence {fn} converges pointwise. Find the limit function. Choose two (2) items only. 1 (a) fn(r)=r"(1−g") (c) fn(2)= (x+n)² x² + n² (b) fn(x) = 1+x" 1+x²n
D. Calculate each of the following limits, or explain why it does not converge. i. limx-1 (x-2)/(x-1)² ii. limx-1 (x²-1)/(x-1) iii. limx-1 |x-1|/ (x-1) iv. limx→ 1+ |x-1| / (x-1)
(b) Let a1 = 1 and an+1 lim, an. Van + 2. Prove that an is convergent and find %3D
3. A particle moving along the x-axis has velocity function v(t) = t²e-t. How far does the particle travel from time t = 0 to t = 5? 4. Consider the sequence {Inn {" n In-1 (a) Write out the first five terms of the sequence (b) Determine whether the sequence converges and if so find its limit