Question M = (M,d) and Y = (Y,d") denote metric spaces. Let (xn) be a sequence in M and let a E M and k ≥ 0. Suppose that d(a,xn+1) ≤ |k|d(a,xn) for all n E N. Let qn = q (n) be a number such that d(a,xn+1) ≤ qn d(a,x₁) for all n E N. Give qn and the range R of values of k for which the sequence (xn) is convergent.

Question M = (M,d) and Y = (Y,d") denote metric spaces. Let (xn) be a sequence in M and let a E M and k ≥ 0. Suppose that d(a,xn+1) ≤ |k|d(a,xn) for all n E N. Let qn = q (n) be a number such that d(a,xn+1) ≤ qn d(a,x₁) for all n E N. Give qn and the range R of values of k for which the sequence (xn) is convergent.

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

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