1).  State whether the statement is true, or give an example to show that it is false. If ∞     n = 1anxn converges, then anxn → 0 as n → ∞. i).  True

Q: 7. Determine the intervals on which f(x)=x³ - 9x² +24x –19 increasing or decreasing; relative maxima…

A: Sorry buddy. As per our guidelines I need to answer only first question if multiple questions are…

Q: Decide whether the following propositions are true or false,providing a short justification for each…

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

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Q: 2. Assume that p > 1. (a) For all n> 1 prove that P-1 (n + 1)P (b) Prove that is convergent. NP <…

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Q: Two of the following statements are true. Which ones? 8 If Σen (1-xo)" converges, then cn (x-xo)"…

A: statement 1 if ∑n=0∞cnx1-x0n converges then ∑n=0∞cnx-x0n converges for all x such that…

Q: Suppose that (an)n20 is a bounded sequence of real numbers. For each natural number n, define fn(x)…

A: We are given a sequence of functions, where {an} is a bounded sequence. We need to establish the…

Q: 1.8.7.) (a) Let an and b, be real or complex numbers. Show that if Yn and bn each converge, then E…

A: Let an and bn be real or complex number

Q: 1+ 1 + 1 V 3 + 1 43 Select one: Qa converges O b. It is inconclusive O c. diverges +...

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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: Exercise 2.1.6: Is the sequence{; Exercise 2.1.7: Let {xn} be a sequence. a) Show that lim xn = 0…

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Q: ? 4. Show that the sequence {fn}, where nx fn(x) = XER 1+n²x²⁹ is point-wise convergent but is not…

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Q: . Find all values of x for which пх" n=1 (n + 1)(2x + 1)" converges absolutely.

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

A: Let us suppose the following sequence xn=1,0,1,0,1,0,..... We have to see that the odd terms are 1…

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: Write TRUE if the statement is always correct and FALSE if otherwise. Prove your answer. (a) If an…

A: The given statement is (a) If an≥0 for all n∈ℕ, then ∑n=1∞an does not converge conditionally.…

Q: Exercise 2.4.3. (a) Show that V2, y2 + v2, V2+ V2 + v2,... converges and find the limit. (b) Does…

A: (a) Given sequence is 2, 2+2, 2+2+2,..... We have to show that the sequence 2, 2+2, 2+2+2,..... is…

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: Show that if lim 0 and E- b, converges then Ena, a, converges too.

A: We have to Prove that series an is convergent.

Q: Exercise 4. 1. Show that if (fn)n converges uniformly on S, then (fn)n converges uniformly on every…

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Q: Let a 1 and define an+1 = V4an -I for n2 1. Show that the sequence {an} converges and find the…

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Q: Suppose (an) is a sequence of positive real numbers such that an+1 Prove that liman (an) even…

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Q: 3η 9.) Ž(-1 2η 9.) Σ n=1

A: Given: ∑n=1∞2nn-13n To Determine: Series is Convergent or Divergent Concept: Root Test:If we have…

Q: Show that a, converges to A iff {a, - A) converges to 0.

A: Answer

Q: Sn/2", l1/2". if n is a prime number otherwise. Let an Does Ea, converge? Give reasons for your…

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Q: Evaluate V4 – x²dx if it converges.

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Q: 2.65 Let fn(x) = xe¬nx² a) Find || fn||sup for all n. b) Determine whether or not fn converges…

A: The given function is fn(x)=xe-nx2 ∀x∈ℝ To find: (a) ∥fn(x)∥Sup ∀n             (b) Whether…

Q: Let X = (xn) be a positive sequence of real numbers. Show that if x₁ + x₂ + + In Yn n then yn is…

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