4. Show that the sequence defined by a=2 An+1- 3-an satisfies () < an < 2...
Suppose that a sequence {Zn} satisfies Izn+1-Znl < 2-n for all n e N. Prove that {z.) is Cauchy. Is this result true under the condition Irn +1-Fml < rt Let xi = 1 and xn +1 = (Zn + 1)/3 for all n e N. Find the first five terms in this sequence. Use induction to show that rn > 1/2 for all n and find the limit N. Prove that this sequence is non-increasing, convergent,
Show that a bounded and monotone sequence converges. Here a
sequence
is called monotone, if it is either monotone increasing, that is
for all
or monotone decreasing, in which case
for all
.
in Sn=1 An+1 > an neN an+1 < an We were unable to transcribe this image
Given that the sequence defined by - 1 2+1 = 5-1 an is increasing and an < 5 for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
Problem 2 Show that if the sequence of numbers (an)n-1 satisfies Inlan) < oo, then the series In ancos(nx) converges uniformly on [0, 27). This means, the partial sums Sn(x) = ) ancos(nx) define a sequence of functions {sn} = that converges uniformly on [0, 271]. Hint: First show that the sequence is Cauchy with respect to || · ||00.
Assume that the sequence defined by a1 = 3 an+1 = 15-2·an is decreasing and an > O for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
Exercises 4.2 ove that the sequence (1 + z/n)"; n = 1, 2, 3,..., converges uni- ly in Iz <R < , for every R. What is the limit? 1, afdos se converge? diverge?
A function is defined as follows: y = X + 6 x² 3x + 1 X<-2 -2<x<3 x > 3 For which x-values is f(x) = 4? Select all that apply 0-2 1 2. 13 e here to search
2. Shade the region of the complex plane defined by <z +4 + 3i : 3 < 3 < 5,2 EC}. Include the appropriate axis labels and any significant points.
A sequence {an , is defined by the following formula. What is the limit of this sequence? do = 3, an= 3an-1-2, for n> 1.
3. (14 pts.) Let the sequence an be defined by ao = -2, a1 = 38 and an = 2an-1 + 15an-2 for all integers n > 2. Prove that for every integer n > 0, an = 4(5") + 2(-3)n+1.