2. Let an be the sequence defined inductively by a₁ = 2 and an+1 = 2 2 an + an (a) Prove by induction that an € [1, 2] for all n € N. (b) Prove that a ≥ 2 for all n € N. (c) Hence prove that the sequence is decreasing. (d) We already know that (an) is bounded below (by (a)) so it follows, by the mono- tone convergence theorem, that (an) converges to some limit L. Show that L> 0 and L² = 2.

Q: 10. Use the definition of the limit of a sequence to prove that (3n+1 lim 2n + 5/ for all positive…

A: we have to use definition of limit of sequence.

Q: 9. Suppose G is the generating function for the sequence 4, 7, 10, 13, 16, 19,..., the find a…

A: The given sequence is 4,7,10,13,16,19,….We need to find the generating function for the sequence of…

Q: 16.) In 2₁ [x], compute ftg (Note: each sequence should only be 0, 1, 2, 3, 4, 5, 6, 7). f(x) =…

A: (16) Given that , fx=x2-3x+2 and gx=2x3-4x+1 find f+g in Z7x.

Q: A sequence of numbers is defined recursively by a 1 =1, a n+1 =√a n +1, n=1,2,3… Show (eg by…

A:

Q: Let =-7 and I+(3, +4) for all integers n2 1. (a) Show that xnS 2 for all n. (b) Show that (x,} Is an…

A: All proofs are easily only  a) proved using induction  b) proved using definition c) find using a…

Q: Determine the limit of the sequence a, u UIS 3 Ve Divergent 4 1 5

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Q: (6) Let {an} be the sequence defined by an+1 = √6an 2, a₁ = 1. Show that {an} is converger and find…

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Q: Prove that if a sequence (an)1 is not bounded, then there is a subsequence (akn)-1 which either…

A:

Q: 81. Happy Thanksgiving!!

A: We have to deduce that an is convergent and find its limit.

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Q: For the sequence {a ,} if a,→ L, and f(x) is a function defined at all a, then f(a ,)→ f(L) Select…

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Q: 35 (b) (5+7k) = k=D1

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Q: 1. Find the limit of the sequence {√3, √√3+√√3, √√3+ √√3 + √3,...}.

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Q: strong induction ICt

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Q: (a) Prove that, for each k E N, b - Ck = ak and b+ Ck = |ak]. (b) Prove that, if ak converges…

A:

Q: 2. Let (2n) and (yn) be Cauchy sequences of real numbers. Define (Tn) to be equivalent to (yn),…

A: Given :  xn and yn be the Cauchy sequences of real numbers. And  (xn) ~ (yn)  if  limn→∞xn-yn = 0 To…

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Q: 1 = 2 and an+1 2 2. Let an be the sequence defined inductively by a1 An + An (a) Prove by induction…

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Q: For a sequence (an), its lower and upper limits are defined by lim inf an = sup inf am = sup inf{an,…

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Q: Suppose real sequences (sn) and (tn) are bounded. (That is, that their ranges are bounded sets.) i.…

A: Bounded sequence and sum of bounded sequences.

Q: Suppose {f,} is a sequence in Banach space B s.t. ||fn - fn+1|| s'/2-). Show that the sequence {f,}…

A: For a vector space X, over a field F, a normed linear space X,· is referred to as Banach space if…

Q: Prove that sequence {an}= {(-1)n} does not have a limit.

A: According to the given information it is required to show that {an}= {(-1)n} has no limit.

Q: 1 Let xn=G, 2+-), nEN be a sequence in R?, then lim x, .A (0,2) .B [0,2] .C (2,3) .D (0,2]

A:

Q: (a) Suppose (k)=1 is a sequence in R" and 1 € R". Prove that the following are equivalent: ● (k)1…

A: 3)a) Suppose xkk∈ℕ  is a  sequence  in ℝn  and l∈ℝn. Assume  that  xkk∈ℕ does not  converge  to l.We…

Q: Let ao, a1, a2, ... be the sequence defined recursively by an = 7(an-1)² if n ≥ 1 ao = 1, Use the…

A: Principle of mathematical induction states that a statement P(n) is true for all natural numbers n…

Q: inf{a,b, : n e N} > (inf{a, : n € N}) × (inf{b, : n E N}). (1) Does the inequality (1) continue to…

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Q: Find a sequence of real numbers {a;} such that lim a, does not n-00 xist, but it has 3 subsequential…

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Q: Prove that there is no sequence of real numbers whose set of accumulation points is exactly Q

A: Answer given below.

Q: 2. Prove the sequence an = n"/n!, n > 1, is %3D (a) increasing; (b) not bounded above (show an 2 n).

A:

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Q: Let (x,) be a convergent sequence of real numbers and let (a,) and (b,) be subsequences of (x,) with…

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Q: 5 Fn+1 3 F,. Compute R6 Define a Sequence R,

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A: Given that, limn→∞a(n)≠0 then we have to find ∑n=1∞a(n) is convergence or divergence or depends.

Q: (b) From part (a), we know that lim an exists. Compute it. n00

A: Given that a1=3 and an+1=12+an for each n≥1 Also, the sequence an is monotonic increasing and…

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A: