Prove that if Σan converges, then Σa also does.

Q: Does (-e)³n! (2n)! (3n)! n=1 converge absolutely, conditionally, or not at all? Justify your answe

A: The series is .To Find:Does the series converge absolutely, conditionally, or not at all?

Q: A) S diverges B) S converges none of these

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Q: 1 1 Зп — 1 3n + 2 n=1

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Q: For each nЄ Z+ let fn : R → R be the function пх fn(2) n|x| +1° (a) Prove that (fn) converges…

A: Given the sequence of functions  defined by .(a)Consider the limit,This is true for every 0≠x∈R.…

Q: Supposean converges conditionally (but not absolutely). Let A+ = {n €N : an ≥ 0} n=0 and A_ = {n € N…

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Q: Find the general solution of the given differential equation. dP+ 2eP - P+ 4t - 2 dt Pe) = Give the…

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Q: If an with AN>O is Convergent, then is Σ TavaN+1 always convergent?

A: Given,  is convergent, where, So,  is.Now, using A.M≥G.M property,We have,

Q: Suppose you run a limit comparison test and determine lanl diverges. In your written work: Nothing,…

A: If series of absolute term diverge than series doesnot converge absolutely.

Q: Which of the following nfnite senes are convergent? Select all that apply. 100 100+50+25 +... %3D 2…

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Q: 43) Suppose that an > 0 for all n and that n=1 anbn necessarily converge? ∞ Σa an converges. Suppose…

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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: 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: Show that X is complete if and only if every absolutely convergent series in X converges.

A: To Show : X is complete if and only if every absolute convergent series in X converges.

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: Exercise 2.5.4: a) Prove that if xn converges, then Σ (x2n+x2n+1) also converges. n=1 n=1 8

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Q: (-1)* k4/3 k=1 a) Converges both absolutely and conditionally b) Converges conditionally c)…

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Q: If a power series converges pointwise on the set A ⊆ R, then it converges uniformly on any compact…

A: Given: K is a compact subset of A. To prove: If a power series converges pointwise on the set A ⊆ R,…

Q: Two sequences of functions fn and gn that converge uniformly but fn In does not converge uniformly.

A: See the attachment

Q: Show if the following question converge or diverges

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Q: Σ (-1)* converges. Vk Determine whether k=1

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Q: i) Let {n} a sequence of non-negative terms that is monotone decreasing and în → 0. If SnX1 X2 + x3…

A: Given that, { xn } is a sequence of non negative terms and this sequence is monotonic decreasing,…

Q: Let {an}1 and {bn}n=1 be sequences. Prove or disprove: If {an – bn}=1 converges to 0 and {an}-1…

A: Note that, if limn→∞an and limn→∞bn exist, then limn→∞an-bn=limn→∞an-limn→∞bn. It is given that,…

Q: Prove the first part of FACT 2, i.e., if an converges and Σ bn diverges, then Σ(an + bn) diverges.…

A: Let Σan converges and Σbn diverges then we have to prove Σ(an+bn) diverges

Q: an2

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Q: (2n)! a"(n!)2 A) diverges if |a| 2² n-1 C) converges if |a| > 2ª

A: Here we have to find the converges or diverges of the given series.

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: 3). True or False. If true prove it, if false give a counter example a) Jf Σax converges absolutely…

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Q: Prove that every conditionally convergent series of real numbers that does not converge absolutely…

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