Prove that {x_n} converges. Need to use info in a) and b)

Q: Part II Write the first five B={ IneN3 then prove sup(B) = l Part III Proue that Pj ajta and b,sb →…

A: a) Supremum of set: Let  S be any subset of real numbers. Smallest upper bound among all upper…

Q: 18. Which of these sequences is/are not bounded? I. a, = (-1)", n> 1 А. II only В. I only С. П only.…

A: Option (C) is correct. II only.

Q: Prove that or quari -1+ X.- ズい 2. X. ntl converged number Ja to 2.

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Q: The seguence of function fn: x→x"+exp (x^) on [o,1] Then [0,1] we have: (fn) on ) neither conuerges…

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Q: ounded funclions on A to h,t converges uniformly

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Q: Let (an) and (bn) be two bounded sequences of positive real numbers. Prove that lim sup(anbn) (lim…

A: Given two bounded sequences of positive real numbers  and .We have to prove that, .To do this, use…

Q: ΣΕΠλη της 9 - 515 +1 _h=1

A: The series is given asAlternating series test -Suppose that we have a series  or  where  for all n.…

Q: Question * Every Cauchy sequence in the Euclidean metric space R" with n a positive integer is…

A: True. I have proved using definition of cauchy and convergent sequence.

Q: ii) 1₁ = √9n²+n- 3n. In = 2.

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Q: A monotone sequence converges if and only if it is bounded.

A: (A) Suppose xn s a monotone convergent sequence.  Since every convergent sequence is bounded,…

Q: Suppose Sanx",Σ bnan converges for all x in (-1,1). Σana" = Σbnan in a subset E of (-1,1) implies…

A: It is given that,  ∑anxn and ∑bnxn are convergent for all x in (-1, 1). Then,  ∑anxn = ∑bnxn in a…

Q: Prove if the following sequences are monotonic and find their limit where possible…

A: According to questionWe have to prove following sequences are monotonic or non-monotonic.

Q: Suppose that ACR is bounded below, and define B = {b € R | b is a lower bound for A}. Prove that sup…

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Q: o prove that knVA=| Test the seuence (ConViordív) of (10) an = (^_-) ) an 15 ら 9n: 3ntl 3n-1 (の+ (7)…

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Q: 32. Find the limit of the series: x (1+x) (+x) 1+x2 A) B) xx X+x C) (1+x D) (1+x) E) x

A: Given series is,

Q: (2) Show that 2003 divides 4007"-1, for all integers n 20.

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Q: Show that in a Banach space convergent an scries is convergent absolutely

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Q: without using the Cauchy Completeness Theorem show that if (sn) and (tn) are both Cauchy sequences,…

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Q: ii) n = √9n²+ n-3n.

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Q: Sp

A: The objective of this question is to understand the formula for 'n choose k', which is a concept in…

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: Assume (v1, v2, v3) and (w1, w2, w3) are independent sequences in the R". Prove that span(v1, v2,…

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Q: -1 6) Show that tan N. converges/di verges. N2 N-1

A: Solution of question as follows

Q: 27. Show that n! nn converges. Hint: Use lim 1148 n (1 + -/-) " n = e.

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Q: rove that then if Ean converges Lim(ant antit t antr) =0,r20 ハ→ 0 O Does imply that Ean Converges?…

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Q: Let (an), (bn) and (cn) be three bounded sequences with an ≤ bn ≤ Cn for each n. Prove that if lim…

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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: Prove by induction that (1+1/5)^n ≥ 1 + n/5 for all n∈N∪{0}.

A: Prove by induction that (1+1/5)^n ≥ 1 + n/5 for all n∈N∪{0}.