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

Q: Exercise 10. Let {an}=1 be a sequence converges to a limit L E R. Prove that any subsequence of…

A: Given a sequence ann=1∞ is converges to a limit L∈ℝ  Let <bnk> be a subsequence of…

Q: |an also converges because 0 If an converges then < lan|, for n = 1, 2, ... n=1 n=1

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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: (ii) Prove that if a sequence Xn converges to a limit I, then any subsequence of Xnalso converges to…

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

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Q: 2* converge? If so, what does it converge to? If not, justify. 7k 1 2 2. Does

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Q: how that fuGe)= x" (1-x") converges bintwise # se G, 1]. Is fn G) Thyenges %3D uni formly?

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Q: + R be defined by fn(x) = an Find the pointwise limit in 1.3 converges uniformly on [–3, 3]?…

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Q: Let {xn} be a bounded sequence of real numbers. Prove that {xn} converges if and only if lim inf xn…

A: Proof

Q: If lim bn pe 0= =1 || En=1 a, converges, then En=1 b, diverges.? True False

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Q: 1 B. Find out if b, where b, = n* sin" 2en -Converges or, -Diverges (Show complete solution using…

A: Given the series is bn=n4sinn12en. Determine whether the series is converges or diverges.

Q: What are all of the values of x such that (-1)"+1 Σ (3)" converges?

A: The ratio test states that: if L &lt; 1 then the series converges absolutely; if L &gt; 1 then the…

Q: Use the definition to show that dx 2x2 converges.

A: We have to show that the given integration converges.…

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

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Q: What are all of the values of x ,such that - (-1) n+A converges? n=1 .1 X26 .2 06

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

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Q: Suppose that anx" converges for x = 5. Must it also converge for x = 4? What about x = -3?

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Q: For sequence of functions {nxe-nx} for x ∈ (0 + 1), what is the uniform norm of fn (x) - f(x) on (0…

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Q: 4n+1 Σ Z 3n – 2 3п — X=1

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Q: If 2 a, converges, then la,| diverges. n=1 n=1 True False

A: Given statement is FALSE. I HAVE given a counter -example.

Q: The interval of converge Use your own wo each step) that yo

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

A: Use Alternating series test, P- series test

Q: Show that the sequence of functions {f.} where f,(x)=: 1+ nx „xER is non-uniformly convergent on…

A: Show that the sequence of functions fn(x), where  fnx=x1+nx2,  x∈ℝ is non-uniformly convergent on…

Q: 3. Determine whether E (-1)" converges absolutely, conditionally or diverges. Explain.

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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: n En=0m! I)Use Cauchy criterion prove that the sequence does not converge uniformly

A: Given: ∑n=0∞xnn!

Q: an2

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Q: If limkoo ak 0 then Eak is convergent.

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Q: Let {xn} be a bounded sequence of real numbers. Prove that {xn} converges if and only if lim inf xn…

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Q: 5. Use the Monotonic Convergence Theorem to show that x, converges. (부) -

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Q: prove convergence or divergeence A) is similar to E by using the Limit Comparison Test tan B)…

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Q: 1 Determine if dr converges or diverges. If convergent, find its value. (In r21

A: Given: ∫0∞1x1+ln x2dx

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

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Q: Suppose that p(æ) > anx" converges on (-1, 1], find the internal of convergence of p(6x – 9). n=0

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Q: Show that the sequence of functions {f.} where f,(x)=nxe is non-uniformly convergent on [0,1]. (by…

A: Test for uniform convergence of sequences: Let fn be a sequence of functions such that it converges…

Q: Prove that or 「+ X。- ズー 2. 71 number J2 Converged to

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Q: Working directly from the e-N definition of convergence (without using any rules for limits), prove…

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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: Use the ratio test to determine if > (-1)". (n!)² (3n)! converges or diverges and justify n=1 your…

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Q: Exercise 2. Use the definition of limit to prove that both of the sequences {+} and {"} converges to…

A: NOTE: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: a) If lim an = 0, then Σ an converges. n→1

A: The divergence test states that for a series ∑n=1∞an, if limn→∞an ≠ 0 , then ∑n=1∞an diverges . That…

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: Given f: [2, )IR, prove that fn (x) = けx" It, converger uni formly to f6)-o.

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Q: 2n+1 4. Use the definition to prove that ) converges.

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Q: 8. Suppose ar > 0 and Does > ak converge or k diverge? Why? %3D

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Q: Find the pointwise limit f(x) for {nxe-nx} for x ∈ (0, +inf)). Does the sequence converge uniformly…

A: Given, fx=nxe-nx , x∈0,∞ Since, limn→∞fx=0 , ∀x x=0, fx=0 and e-nx→0 as n→∞ Let, fx converges…

Q: 8) Show that the sequence 12, V12 + vi2, v12 Is bounded and monotonic. Then find its limit. 12 V12…

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Q: 04: A) Discuss the convergence of f,(x)=x+- -.on A [0,0).

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