3. Examples. Justify your answers briefly. (a) Let A be the family of functions defined everywhere on R so that lim f(x) converges. Show that A is an algebra of functions. (b) Give an example of a set with a finite number of points that is not se- quentially compact, or else explain why no such set exists. 00 (c) Give an example of series an and > bn that are divergent, but so n=0 n=0 00 that an converges. n=0 n=0

Q: If lim an 1 then n00 Σ an n=1 converges O diverges O may or may not converge

A:

Q: 1. Give an example of a sequence {an} where lim(an+2 – an) = 0 but {an} is divergent.

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Q: True of False: If the geometric sequence {cr"}n2o (c # 0,r 0) converges, then the associated…

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Q: Which of the followings are true? I. If 1 an is convergent then lim->00 am = 0. II. If lim-00 n = 0…

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Q: 6. Find the radius of convergence and the interval of convergence for E-0¹ (x-2)".

A: 6. The given power series is     ∑n=0∞12nx-2n We need to find the radius of convergence and the…

Q: The series Σ 6( − 1)*0.2-1.9/k○ diverges converges. k=0

A: We use ratio test to find the desired result.

Q: Let fn(x) be the following sequence (not series) of functions n2 fn(x) : (x2 + 7x + 6)n² +n (1) Is…

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Q: Given that lim n2(x- 5)" converges by the Ratio test. If x€ Z then the numerical n- 00 value of x is…

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Q: whether the sequence an = 1+ converges and, if it does, find the limit. ) diverges 3 O converges to…

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Q: السؤال 6 If lim=c>0 , then b, . a, both convergent and both divergent. ylgn O ihi O

A: True

Q: What can you conclude about the convergence or divergence of ∑an?

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Q: Given that lim n²(x– 5)" converges by the Ratio test. If x € Z then the numerical n- 00 value of x…

A: This is a problem of power series.

Q: B. Prove that limx → c f(x) = o if and only if for every sequence aŋ that is in the domain of f and…

A: Consider a function fx. Suppose that limx→cf(x)=∞. That is for all N>0, there exists a δ>0,…

Q: Use series to evaluate the limits

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Q: 13.I. Suppose that (xn) is a convergent sequence of points which lies, together with its limit x, in…

A:

Q: In the given question as follows, determine whether the statement is true or false. If it is false,…

A: Given statement, If limn→∞ an=0  then ∑n=1∞an converges

Q: 1.1 Let {fn (2)} = {1 NI +nz ,z ER and 2 > 0. 1.1.1 Find the pointwise limit of the sequence {fn(x)}…

A: To Examine: The series of function Sn(x)=∑k=1∞xk-1 converges uniformly on [-a,a], a∈(0,1) or not.

Q: V. Let f(x) be a differentiable function such that for all x > 1, f(x) • f(x) > 1, f'(x) (-1)" f(n)…

A: Solution is given below:

Q: 1.1. For r € (0, 1] (1) = f(r), wher

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Q: Σ (-1)*2* Use the Ratio Test to determine whether converges absolutely diverges. or k! k=1

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Q: Given that lim n2(x-5)" converges by the Ratio test. If xEZ then the numerical 1- 00 value of x is…

A:

Q: 3. Let f(x) — х3 — Зх — 2 (a) Define the Newton-Raphson sequence {xn} for the given f(x) (b) Start…

A: Given that function f(x) = x3 - 3x -2 Suppose that f(x) is differentable function and we know the…

Q: The integral tests says that if an=f(n), then the series 2 an is convergent if and only n =1 if the…

A: for integral test three conditions must be satisfied for the function i.e.  continuous, positive and…

Q: (b) Prove that if a sequence converges, then its limit is unique. That is, prove that if limn→0 Sn =…

A: We have to give the proof of the second part i.e., part (b) of problem 7.4.2. We have to show that…

Q: 11.J. If X and e quals a number less than 1, then lim X = with ) 0, then 0 < xn < Ar" for…

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Q: V. Let f(x) be a differentiable function such that for all æ > 1, • f(x) > 1, • f'(x) < f(x) and lim…

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Q: Suppose x1 : 1/2 and xn+1 := xn2. Show that {xn} converges and find lim xn.

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Q: 4. Consider the folowing sequence of functions. 5+ nx? fn(x) = , x E [0, 1] 5+ nx a) Find the…

A: (4.) We will find the all step of the function.

Q: Σ 1 a, where a, = Consider the series (In n)º n=4 Compute the limit: lim ya, = (Enter "infinity" or…

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Q: Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is…

A: Given that: ∑n=1∞enn!

Q: 4. Consider the sequence of functions {fn}=2 where fn: [a, b] R defined for n > 2 by if 0 < x <- fn…

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Q: A) If (xn) is a decreasing and bounded sequence. Prove that (xn) is convergent with lim xn = inf {xn…

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Q: 23 = 24 = a5 = In(n) lim = -0 n+ 1 Enter DNE if limit Does Not Exist.) Does the sequence converge…

A: The given sequence isln(n)n+1n=1∞ and an=ln(n)n+1.

Q: Question 3. Let g, = nx1/n.2/a] and g = 0. Show that 9 lim Gn. Does the sequence (gn) converge…

A: Let gn=nℵ1n, 2n and g=0. Show that ∫02g ≠limn→∞∫02gn. The characteristic function ℵA is defined as…

Q: (b) Let X and explain whether the sequence X = (Xn) converges or diverges. (xn) be defined by xn =…

A: We have to solve 4.b

Q: V. Let f(x) be a differentiable function such that for all æ > 1, • f(x) > 1, • f'(x) < f(x) and lim…

A:

Q: I have no idea how to approach this question

A: Known fact:

Q: What can you conclude about the convergence or divergence of an? (converges, diverges, or…

A:

Q: 8. True or False: If lim an = 0, then an converges. %3D n=0 (a) True (b) False

A: Solved below

Q: If lim=c>0, then both convergent and both divergent. ylgn O Ihi O

A: If limn→∞anbn=c&gt;0, then ∑bn,  ∑an both convergent and both divergent. (a) True (b) false

Q: 7. Suppose we know that lim an 0. What, if anything, can we conclude about the convergence or…

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Q: 5. Find the radius of convergence and the interval of convergence for E- (x – 2)" n%3D1

A: To find the interval and radius of convergence.

Q: n! converges. Hint: Use lim Show that = e. пh п-1

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Q: 13.1. Suppose that (xn) is a convergent sequence of points which lies, together with its limit x, in…

A:

Q: Q(A). Let {fn(x)}=1 = 1+ (x – 2)* , be a sequence of functions | n=1 defined over [2,3]. Show that:…

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Q: 17. Find the interval of convergence for E(-1)"-1÷) x'

A: To find the interval of convergence.

Q: 7. Suppose (fn) is a sequence of continuous functions on an interval I that converges uniformly on I…

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Q: True or False? Prove your answer! a) Suppose the sequence (xn) does not converge to 0. Then, for…

A: a) Given statement is   "Suppose the sequence xn does not converge to 0. Then for every ∈&gt;0,…

Q: 65. Let S = E cn! where c is a constant. Σ n" n=1 (a) Prove that S converges absolutely if |e| e.…

A: