Use the Bolzano-Weierstrass Theorem to show that (Xn) = (1, 3, 5, 7, 1, 3, 5, 7, 1, 3,5, 7, . ....) contains a convergent subsequence.

Q: How to prove {sin(n)} is a bounded seuqence that does not converge?

A: To prove xn=sinn is bounded. By definition, a sequence xn is bounded if it is both bounded above and…

Q: Show that x, converges to x.

A: Given that X be a metric space . Let xn be the sequence in a metric space X. Suppose that xn has a…

Q: Show that a sequence of Riemann-Integrable functions fn(x) converges uniformly to Riemann-integrable…

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Q: oblem 3. Use the integral test to determine whether Y converges or diverges.

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Q: Suppose fn:A→ Ris uniformly continuous on A, Vn E N and fn →f converges uniformly on A. Question:…

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Q: Let f be a uniformly continuous function on (a, b) and suppose that {xn} is a Cauchy sequence in…

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Q: How to determine nln(n)/(n+1)^3 is convergent or divergent?

A: To determine the sum of the series nln(n)/(n+1)^3 is convergent or divergent. Now, Apply the…

Q: 7N 8N 3) Show that converges/diverges.

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Q: Decide whether the following proposition is true or false. Provide a justification for your answer:…

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Q: Suppose that the sequence of functions {fn} and the function f are integrable on [a, b] and that…

A: Given that,  The functions fn and f are integrable on a,b. fn converges to f on uniformly a,b. an…

Q: 4. Given the FOl owing conditions, give an example and explain your an swer. 8 a. Cn is convergent…

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Q: Let fn(x) = n²x²(1 – x) on [0, 1]. On what closed subset, F C [0, 1], does the sequence {fn}…

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Q: if Xn is bounded, is Xn  cauchy ?  is Xn convergent?

A: Solution

Q: Sequences {xn} and {yn} such that they are both convergent and {xn+yn} maps to 5

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Q: State why the following intergral is improper and find the value if it is convergent: 2r (z² – 4) "

A: The given integral is I=∫262xx2-423dx The integral is improper as the integrand is unbounded at x=2.

Q: Suppose that we observe that X1, X2, . . . , Xn  are iid∼ U(0, 1). Show that X(1) converges in…

A: A sequence of random variables Xn is said to converge in probability, denoted as Xn→pX, if…

Q: (b) Give an example to show that the product (fngn) may not converge uniformly.

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Q: Let fn(x) = x¹(1-x), r [0, 1] and n € N. Show that fn = 0 on [0, 1] but that {f} is not uniformly…

A: Given fn(x) =xn(1-x),             x∈[0,1] and n∈N To prove: {f'n} is not uniformly convergent.

Q: Test the convergency o f Stan ("

A: According to the given information, it is required to test the convergence.

Q: Prove that sina de converges but not absolutely for 0 < p < 1. XP

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Q: Corver Is absdutely contergent, Conditionally Convergent, on divergent

A: Use Alternating series test, P- series test

Q: A sequence of functions that converges pointwise on [0, ∞0) but does not converge uniformly [0, 0).…

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Q: Let (hn),(tn) be sequences of bounded functions on A that converges uniformly on A to h,t…

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Q: 4. Suppose fn:A→ Ris uniformly continuous on A , Vn e N and fn →f converges uniformly on A.…

A: Given: fn: A→ℝ is uniformly continuous on A ∀n∈ℕ To prove: fn→f converges uniformly on A  fn→f…

Q: If {ah} is inc reasing and bounded above, then it is convergent. صواب ihi

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Q: 2n-1 3n+2n+1 n=1

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Q: If {ah} is increasing and bounded below, then it is convergent صواب ihi

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Q: Let fn(x) x + n a. Show that fn(x) converges pointwise [0, 0). on

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Q: Find the smallest positive integer n so that S, approximates the sum of the convergent ser to four…

A: Here we have to find the smallest positive integer n so so the Sn, approximates the sum of the…

Q: Determine whether the statement is true or false. dr is convergent.

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Q: Determine if the given is convergent or divergent. Indicate what test is used In n 1. n=1

A: To determine that the following is convergent or divergent  ln2nnn=1∞ ∑n=2∞ln2nn Root test: If…

Q: 3- a- By definition show that (x,) = (+3) converges to 3. b- Does (+3) bounded, Cauchy? Show by part…

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Q: show : ΣΣ - a"d, (x - a)" | n=0 m=0 is convergent,

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Q: (a) Determine whether {an} is convergent.

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Q: Use the Raho Test to determine whether thc serics is convergen Or divergent oo(-3 n 0

A: Here, the general term of series  an is shown on board. Forstly, we calculate next term and…

Q: Suppose that p(x)=an" converges on (-1, 1], find the interval of convergen n=0 8 left end included…

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Q: Let fn(x) x +n b. Use the Lemma for non-uniform convergence to show that the sequence does not…

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Q: If \( \{a_n\}\)  is decreasing and bounded below then it is convergent.  true or false?

A: Monotone Convergence Theorem : Every bounded monotone sequence is convergent.

Q: /5 too dr Vx-1

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Q: .Use the Monotone Convergence Theorem to show that the (2n)! {a)" is convergent.

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Q: If Ea, converges absolutely, prove that Ea,² converges.

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Q: chooes led R- to ) and P= then OR convergentan P divergent O both Rand Par convergent O R…

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Q: 3D (2Vn +1)n

A: The given series is,∑n=1∞2n2+1n

Q: b) Use the Comparison Theorem with f(x) = - to show whether , g(x)dx is convergent or divergent.

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Q: Determine if the following is convergent or divergent: a dr

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Q: Use the integral test to determine if -2 converges or diverges. Justify your en n n=1

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Q: 1.4 Does (fa(r)} in 1.3 converges uniformly on |-3,3]? Explain.

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Q: cos" (;) is absolutely convergent, conditionally convergent, or divergent. en?

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