3] Let (xn) be bounded (not necessarily convergent) sequence, and assume that lim(yn) = 0. Show that lim(x„yn) = 0.

Q: 1. Give an example of a series such that > an converges but > an| diverges.

A: To give an example for: ∑an converges but ∑ an diverges.

Q: 21

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Q: Show that a convergent sequence is upper limited. (Show that  limn→∞ xn = L entails |xn| ≤ K, where…

A: Answer

Q: Let {xn} be a bounded sequence on R. Show that Xn lim = 0. - n→+o n

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Q: 3. Sn = + + ...+ Show that the {sn} sequence given in general terms converges %3D

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Q: 16) Determine the convergence or divergence of the sequence with the given nth term. If the sequence…

A: We determine the convergence or divergence of the given sequence by sandwich theorem. The sandwich…

Q: Determine whether the sequence 2ñ sin converges or diverges. If it con its limit. 27 1 1 2 diverges

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Q: 22: Establish the divergence of the sequence x, =(-1)"

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Q: 3 and Xn %3D 4-Xn-1 converges. %3D Prove that the sequence defined by x|

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Q: converges and find th Yn -2- Use the Monotone Convergence Theorem to show that the sequence defined…

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Q: Let {xn} be a bounded sequence on R. Show that Xn lim = 0. - n→+∞ n

A: follow next step

Q: Assume (fn) and (gn) are uniformly convergent sequences of functions. (b) Give an example to show…

A: Given : Assume (fn) and (gn) are uniformly convergent sequence of functions.

Q: Examine the convergence of the sequence an=sin2n/2n whose general term is (an) Find the limit with…

A: Given sequence is an=sin2n2n. We have to check convergence of the given sequence and then find the…

Q: Let {an} be a monotonic sequence such that an ≤ 1. Discuss the convergence of {an}. When {an}…

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Q: (п +5)(п — 3) 1 | converges. If so, find its limit. n-1 Determine whether the sequence 4n?

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Q: 13.I. Suppose that (xn) is a convergent sequence of points which lies, together with its limit x, in…

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Q: Find the limit of the sequence if it converges; otherwise indicate divergence. an in 1 0. 7.…

A: an=7n.nn

Q: ii) Prove that if a sequence XnXn converges to a limit l, then any subsequence of XnXnalso converges…

A: Just use definition of convergence.

Q: 1. Determine whether the sequence converges or diverges. If it converges, find the limit. a, = (2n)…

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Q: 3Vn + 13 Determine whether the sequence an converges or diverges. If it converges, find the limit.…

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Q: A) If (xn) is a decreasing and bounded sequence. Prove that (xn) is convergent with lim xn = inf {xn…

A: Multipart: (A)  We need to prove that , if xn is decreasing bounded sequence then , xn is convergent…

Q: 11. b) Determine whether the sequence an = 2- cos(n) is convergent or divergent.

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Q: (b) Prove that if a sequence converges, then its limit is unique. That is, prove that if limn→0 Sn =…

A: We have to give the proof of the second part i.e., part (b) of problem 7.4.2. We have to show that…

Q: 3.2.6 Theorem If X (x) then a < lim(x) b. is a convergent sequence and if a S X5bfor all n e N,

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Q: 7. Determine whether the sequence, n 1+4n² 1+n² > converges or diverges. If it converges, find its…

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Q: (b) Give an example of divergent sequences {a,} and {b,} such that {a, + bn} converges.

A: To give an example The sequences {an} and {bn} are divergent sequence such that{an+bn} converges.

Q: A) If (xn) is a decreasing and bounded sequence. Prove that (xn) is convergent with lim Xn inf {xn :…

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Q: Consider the sequence An , whose nth term is given by An = 2(1 + 0.025/n)^n where n is the number of…

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Q: Let (Xn)n and (Yn)n be two convergent sequences with limits x∗ and y∗. Prove that (a) xn + yn is a…

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Q: A) If (xn) is a decreasing and bounded sequence. Prove that (xn) is convergent with lim xn = inf {xn…

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Q: Analyse the convergence of the sequence {fn} where fu(x) = on R. 1+ na²

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Q: converges for all * > 0 because lim a - In n 0, for all * > 0- n=2 True False

A: Need to check given statement is true or false.

Q: Find the limit of the sequence if it converges; otherwise indicate divergence. tan-1 n 1) an

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Q: 4. Determine if the sequence (sin²n+ 2n converges or diverges. Find the limit if it converges.

A: we have to identify the given sequence is convergent or divergent if convergent then find it's limit

Q: b) an = (-1)"/n (-1)" Vn c) an

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Q: 4. Suppose that lim s, =0 and suppose that (t, ) is a bounded sequence. Show that lim s,t, = 0 %3D

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Q: prove that if X =CXn) is bounded increasing sequence of real ConVergent and Lim (xu) - Sup [An ineN]…

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Q: 3. Determine whether the sequence an = limit. x + 3 In x converges or diverges. If it converges,…

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Q: sin n 2 T Determine the convergence of the sequence a %3D If it converges, find lim an- Paragraph BI…

A: NOTE: Refresh your page if you can't see any equations. . find the limit of the sequence

Q: 28. yn = ne/n

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Q: sin? a = 2"

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Q: (a) Give an example of a convergent sequence (x,) of positive numbers wich lim(x:/x,) = 1. (b) Give…

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Q: 1+ 2n² Xn {Tn} 1– 3n2 - Prove that the sequence where is bounded.

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Q: Use series to evaluate the limit. X-In(1 +x) lim

A: Find your answer below

Q: Let {xn} be a bounded sequence on R. Show that Xn lim = 0. n→+∞ N

A: Given sequence {xn} is bounded we want to show that  limn→+∞xnn=0 we  use the limit limn→∞1n=0

Q: 4. Suppose that lim r, = 0. If (yn) is a bounded sequence, prove that lim In Yn = 0.

A: Bounded sequence: If there is a positive number M such that |Xn|≤M for every n∈N, the sequence xn is…

Q: A) If (xn) is a decreasing and bounded sequence. Prove that (x,n) is convergent with lim xn inf {xn…

A: Multipart :    xn  is decreasing and bounded sequence. To show that , xn is convergent and limn→∞ xn…

Q: Find out if the sequence is convergent or divergent. If convergent, find limit. lim x→∞ cos^2x   /…

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Q: EXE CISES Prove that every uniformly convergent sequence of bounded functions is uni- formly…

A: Use the definition of sequence of functions to solve this problem, as mentioned only first, in next…