Let (Xn)n and (Yn)n be two convergent sequences with limits x∗ and y∗. Prove that (a) xn + yn is a convergent sequence with the limit x∗ + y∗. (b) xnyn is a convergent sequence with the limit x∗y∗.

Q: Prove that if the sequence (xn)n is convergent, then its limit is unique.

A:

Q: Let {an}=1 be a convergent sequence such that a1 = V5 and an+1 = V5 a,n. Compute lim an.

A: The sequence an is convergenta1=5 and an+1=5 an

Q: use the appropriate limit laws and theorems to determine the limit of the sequence orshow that it…

A:

Q: Let (a,) be an unbounded sequence and (b,) be a convergent sequence. Prove that the sequence (a,-b)…

A:

Q: Determine the limit of the sequence a, (-1)* 1 1 2 2 - 1 Divergent

A:

Q: For the sequence { 4^(-n) } a) determine whether or not is convergent b) if it is convergent,…

A: The nth term of the given sequence is given by bn=4-n. If limn→∞bn exists, the sequence bn is…

Q: use the appropriate limit laws and theorems to determine the limit of the sequence orshow that it…

A:

Q: Assume that the recursively defined sequence converges and find its limit. 15 a, = 1, an + 1-2+ an…

A:

Q: a1 =-1, an+1= An +6 An +2

A:

Q: Let {an}n=1 be a convergent sequence such that a1 = v5 and an+1= /5an Compute lim an

A:

Q: Use the definition of the limit of a sequence to prove that (-1)"n' lim = 0. n2 + 1

A: To prove limn→0(-1)nnn2+1=0

Q: Prove that if {xn} is a sequence such that 1+ n |æn| < for all n E N, 1+ n2 then the sequence {xn}…

A:

Q: 2. a. Prove using the formal definition of a limit that the following sequence converges: -n3 + 2n –…

A: This is a problem of sequence.

Q: Prove the sequence converges and find its limit.

A:

Q: Assume that the recursively defined sequence converges and find its limit. a1 = - 2, an +1 = /14 +…

A:

Q: Ρτοve that the Sequence Converges and find i ts limit.

A:

Q: use the appropriate limit laws and theorems to determine the limit of the sequence orshow that it…

A:

Q: Assume that the sequence converges and find its limit. 72 a 1 = 5, a n+ 1- I+ an

A: Given that the sequence converges and the sequence is defined as a1=5, an+1=721+an. Let L be the…

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: Assume that the sequence converges and find its limit. a1= 3, a n+ 1 = V8 + 2an

A:

Q: Let the sequence {fn(x)} be defined on [0, 1] by n – n²x; if x E (0, ) = (x)"f 0; otherwise Find the…

A: we have sequence of function fnx=n-n2x ; if x∈0,1n0;           otherwise pointwise limit of sequence…

Q: ] Let {an} be a convergent sequence of real numbers and let lim a, = a. n00 (a) Prove that {a,} is…

A:

Q: Please help

A: The given recursively defined sequence is

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: a. In a metric Space (X, d) if (xn) and (yn) are two sequences in X such that : X - x€ X and yn → y…

A: a) In a metric space (X,d) Let (xn) and (yn) be two sequences such that xn -&gt; x and yn -&gt; y…

Q: Let {x„}" and {ym}*=1 be convergent sequences with limits x and y, respectively. Suppose that all…

A:

Q: 10. Determine whether the sequence converges or diverges. If it converges, find the limit. (»)…

A:

Q: Determine whether the sequence defined by 1/ In(n) an = is convergent or divergent. If the sequence…

A:

Q: Consider the iteratively defined sequence (x,), where x41 = 3 - 2 and x = 3. Use the Monotone…

A: Monotone convergence theorem: If a sequence monotone and bounded then it is convergent.   If the…

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

A:

Q: 1. Consider the sequence Xn  = √n + 1 − √n, n ≥ 1. Prove that (xn)n is convergent. Find its limit.

A:

Q: 20

A:

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

A:

Q: Prove using the ϵ−n0 definition that the sequence Xn=(9−7n)/(8−13n) converges, and find its limit.

A:

Q: use the appropriate limit laws and theorems to determine the limit of the sequence orshow that it…

A:

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

A:

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

A:

Q: Let {an}=1 be a convergent sequence such that a1 = V5 and an+1 V5 an. Compute lim an.

A:

Q: If {xn} is a bounded sequence on R. Prove: lim n→+∞

A: Given  sequence xn is bounded  we want  to show that  limn→∞xnn=0

Q: Let x1 = 1 and define xk+1 = sqrt(2xk) where k is a natural number. Prove that the sequence {xk} for…

A: Given: x1=1, xk+1=2xk So, x2=2x1=2&gt;x1 x3=2x2=22&gt;x2 On generalizing, we get…

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

A:

Q: Prove that the limit of the convergent sequence is unique.

A:

Q: Determine whether the given sequence is increasing, decreasing, or not monotonic; bounded below,…

A:

Q: n (1+)" 66. An n2

A:

Q: Prove that any convergent sequence {xn}=1 is bounded.

A: We want to prove every convergent sequence is bounded.

Q: Use the Theorem of Sequence Defined by a Function to determine the limit of the sequence or state…

A:

Q: 40. d, = Vn + 3 – yn

A:

Q: Let {a,} be a convergent sequence of real numbers and let lim a, = a. (a) Prove that {a,} is…

A:

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: 2n – 7 Prove that the sequence with nth u, = (a) is monotonic increasing, (b) is bounded above, (c)…

A: Given a sequence un = 2n-73n+2 a) It is monotonically increasing if  un≤un+1 ∀n∈ℕ We need to show…