1. (a) Let an→ 2. Prove from first principles (i.e. give a direct e-N proof) that a → 4. (b) Let an → 4. Prove that (an) has a subsequence which converges either to 2 or -2. (c) Let a →→ 4. Prove from first principles (i.e. give a direct e-N proof) that |an| → 2.

Q: 3. Use the Test for Divergence to explain why if En-0 an converges, then En=0 am must diverge.

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Q: .3 Let fn : [-3,3] → R be defined by fn(x) = . Find the pointwise limit of fn(x). 2n+1' .4 Does…

A: Given,         fn : [-3 ,3] -> R be defined by ,                             fn(x) = x(2n)/{x(2n)…

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Q: n/Jn-1 35. Suppose x is an accumulation point of (a,: n E J). Show that there is a subsequence of…

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Q: 6. Suppose that the sequence {an} converges to a and that a > 0. Show that there is an index N such…

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Q: uppose that the sn satisfies both limn→∞ s2n = 3 and limn→∞ s2n+1 = 3. (That is, the sequence given…

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Q: True or False? Prove your answer! If f : R → R is continuous and (xn) is a sequence such that…

A: If f : R → R is continuous and (xn) is a sequence such that (f(xn)) converges to f(c), then (xn)…

Q: 65. Suppose that the coefficients of F(x) = anx" are periodic; that is, for some whole number M > 0,…

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Q: a. Show that if two power series E-oa„x" and E-o b,x" are convergent and equal for all values of x…

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Q: Assume you have given a sequence c,, with non-zero values (cn #0 for n = 1,2,...) that fulfils the…

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Q: 1. (a) Let an → 2. Prove from first principles (i.e. give a direct e-N proof) that a → 4. (b) Let a?…

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Q: a. 0 a.c .Q. Consider (R, M(R), µ). For x € [2,3], consider the sequence of functions tan ((x –…

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Q: re 1. (a) Let fn(x) = (1-x)x". Prove that the sequence converges point-wise on [0,1]. Find the limit…

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Q: 4. Let TN → N be an injective function. Prove that if {n}1 converges to x, then {*T(n)} also…

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Q: 2. Suppose that {an}1 converges to 1 and {bn}1 converges to 2. For n € N, define an: if n is odd; Cu…

A: Please see the below picture for detailed solution.

Q: 2. Let fn : [1, 2] → R be a sequence of functions defined by fn(x) = x*. Find the pointwise limit of…

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Q: .Q. Consider (R, M(R), µ). For x € [2,3], consider the sequence of functions tan ((х — 2)"п) - fn(x)…

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Q: x +x2+..+ xn Let 0 is convergent, show that is convergent or divergent (2)Let is convergent, show…

A: (a) Given that an=x1+x2+...+xnn, bn=x1x2...xn1n It is known that bn≤an Since an is convergent. By…

Q: Prove by contradiction the scquence {(-1)"+} is divergent.

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Q: 2n Let ɑn en O {an} is divergent. O {an} is convergent. 2 {an} converges to e O {an} converges to 0.

A: A detailed solution is given below.

Q: 6. S= 00 (-1)^2n 3" n=0 =o =0--+ 2 4 39 6 8 + 27 81 (a) Show steps and find lim antl 11-0 an +...…

A: Given: S=∑n=0∞-1n 2n3n=0-23+49-627+881+...To Find: a To find limn→∞ an+1an.b Based  on part a, to…

Q: Prove: if Σ1 fn is convergent on D, then limn→∞ fn(x) = 0 for each x = D. n=1

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Q: For eksample Xn = n² (n+1)! investigate whether the serles Σ 8 n=1 Xn convergent Or divergent. Prove…

A: We will use ratio test to check whether given series is convergent or not.

Q: Suppose (xn) does not converge to 0. Then there exists e>0 such that all except for finitely many…

A: Consider the sequence xn=1,0,1,0,1,0,....... Here the odd terms are 1 and the even terms are 0.…

Q: Prove that V n E N, 2n

A: Since you have posted multiple questions but according to guidelines we will solve first question…

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Q: 9. Let g RR be a bounded, differentiable function whose derivative g': R → R is also bounded. For…

A: Approach to solving the question: Detailed explanation: Examples: Key references:Introduction to…

Q: (iii) 2 an must converge if lim an 0. n =0 bn (iv) 2 bn must diverge if b, > 0 and lim = 4. n->0…

A: The solution are next step is

Q: Let ƒ : [−1,1] → R be a continuous function. Prove that the sequence (n) defined inductively by Xn 1…

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Q: O (*+n)* Given fafx)= x*+n?' find NeN such that fu (x) converges to f(x) =1 Vx e [,-) a) E c) d)…

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Q: by Consider the sequence of functions (fn) defined on the interval [0, 1] given n²x², 0≤x≤ 1/n 1,…

A: As per policy the first three subparts are answered.   The given sequence of functions is…

Q: Exercise 2. Determine whether the sequence of functions (fn)n converges uniformly on S. 1 fn(x) = S…

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…