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Q: Prove that if the sequence {an} is divergent and the sequence {bn} is convergent, then the sequence…

A: Given: The objective is to prove that if the sequence {an} is divergent and the seqence {bn} is…

Q: (6) Lettan }n=obe a sequence. Prove that if an n=o is decreasing and not bounded below, then it is…

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Q: Show theit the fenias +- (AtI)(28+1) (P+1)(2p41)(38+1) (uhese a>0, B>0) conveages if B>a amd…

A: Ratio test If we have series ∑ann=1∞ Then we will find the limit l = limn→∞an+1an If l<1 , then…

Q: a_k=a_(k−1)+4k + 1, Vk ≥ 1, k € Z, a_0=0 Find explicit formula for a n use iteration, sum of…

A: Given ak=ak-1+4k+1,forall k>=1 k∈Z,a0=0 To find an explicit formula for an, we will use iteration…

Q: 3. (Cauchy condensation principle) positive numbers. do I xn n=1 Show that converges бирроге (хия й…

A: This question is about sequence and series.

Q: O4l determine if the geometric series converges o diverges o If seri es converg es, find its sum 3.…

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Q: Determine whether the sequence is monotonic and whether it is bounded. sinyn an

A: It is required to check whether the given sequence is monotonic and bounded.

Q: Sequences {xn} and {yn} such that they are both convergent and {xn+yn} maps to 5

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Q: Using any technique from this class that you like, show that if (sn) 1s a Cauchy sequence of…

A: Given that an is a cauchy sequence. The objective is to show that an is a cauchy sequence. Then the…

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A: "Since you have posted multiple questions, we will provide the solution only to the first question…

Q: dcreasing sequence. and the limit for it =0

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Q: A sequence {an} is called decreasing an S an+1 for all n E N. ulgn

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Q: an n In n en (domain n ≥ 1)

A: Given that an=nln(n)en We have to find it's limit when n tends to infinity.

Q: 1S niformly ) Let {n} be a sequence of real numbers and x E R. Suppose there exists M = N such that…

A: To prove that the sequence {xn​} converges to x, we need to show that for any given ϵ>0, there…

Q: If an diverges and 0 (-1)"an converges. n=1 B)>(-1)"b, converges. n=1 c) 2(-1)"bn diverges. n=1 D b,…

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Q: 10 Show that (S) converges to a continuous function, but not uniformly. Use the series E f, to show…

A: Solution: The function fnx is defined as follows: fnx=0              x<1n+1sin2πx     1n+1≤x≤1n0…

Q: 4+6 n i) The sequence { )is bounded by 6 n <ns ii) The 20 th. element of this sequence is

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Q: an n In n en (domain n ≥ 1)

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Q: g) an = In(n+ 1) – In(n) (Hint: simplify the logarithms) (-3)" h) an n! i) an = Vn + 2 – V2n

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Q: Question: Let {an} be a sequence and L  is any real number. Prove that {an} =1 converges to L  if…

A: Consider a sequence, an and L be any real number. To prove that, an converges to L if and only if…

Q: 2. Suppose that {an} is a bounded sequence and En an diverges. Prove that R = 1 is the radius of…

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Q: Prove {an} is decreasing for n >>1, if ao=1 an

A: A sequence {an} is said to be decreasing when we have  an+1-an&lt;0

Q: 4+4 n i) The sequence { ") is bounded by 3 n ii) The 11 th. element of this sequence is

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Q: Question 2* (Cauchy criterion) Recall that a sequence s, is a Cauchy sequence if for all > 0, there…

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Q: Converges absolutely Burame

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Q: Question 8' Given the following general terms an, determine whether the corresponding sequence…

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Q: 2. SUBSEQUENCES (a) Let (r) be a sequence and let (n) be a subsequence. Show that if (n) is monotone…

A: We prove the the given problem.

Q: Which step of the following of the proof of the fact that "if a -→ a, as n 0o, then JaJal" not true?…

A: For 1st the B option is correct .

Q: 15 Let (a,) be a sequence of real numbers. Which of the following statements about (a,) is true? O…

A: Which of the following is true

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Q: Use Cauchy Criterion to prove that the sequence {sn} converges where Sn + Sn-1 Sn+1 n > 1, 2 and…

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Q: Suppose SELECT ALL A) sometimes converges depending on the sequence, a(n) B) lim a(n) # 0 never…

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

Q: Suppose f: D→ R with ao the only accumulation point of D. If {an}1 is a Cauchy sequence with an ED…

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