Let {xn} be a sequence of real numbers. Let yn be the average of the first n-terms: x1+ x2 + Yn + Xn n Prove that if {Xn} converges to x, then {Yn} also converges to xo. (Hint: using the definition of convergence, choose a large threshold N after which |xn – x| is as small as we want, then choose n large enough such that the |x; – x|/n is very small for 1 < i< N.)

Q: Let the sequence (xn) defined by 1 X1 = 3, Xn+1 4-xn (a) Show that 0 < xn < 3 for all n E N (Hint:…

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Q: Prove that if {sn} converges to L and L &gt; 0, then there exists a number N such that sn &gt; 0 for…

A: Let sn be a convergent sequence and converges to L&gt;0. We have to prove that sn&gt;0.

Q: Let sn be a bounded sequence, and let tn converge to L. i. Show that if L = 0, then the product sn ⋅…

A: We know that the definition of convergent sequence and bounded sequence, A sequence an is said to be…

Q: Consider the sequence {fn(x)}, where sin(nx) fn(x) = 1+ nx Use the uniform norm to show that the…

A: Given: fn(x)=sinnx1+nx To show: fn(x) is uniformly converge on [1, ∞).

Q: Mark all answers that are correct (there might be more than one correct answers). O Every bounded…

A: only option 1 is true.

Q: For a convergent real sequence sn and a real number a, show that if sn ≥ a for all but finitely many…

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Q: 2. Assume that p > 1. (a) For all n> 1 prove that P-1 (n + 1)P (b) Prove that is convergent. NP <…

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Q: Suppose that (an)n20 is a bounded sequence of real numbers. For each natural number n, define fn(x)…

A: We are given a sequence of functions, where {an} is a bounded sequence. We need to establish the…

Q: Help for 2a,2b,2c.

A: Given,

Q: . i. For a convergent real sequence sn and a real number a, show that if sn ≥ a for all but finitely…

A: Given: Convergengent real sequence sn and a real number a. To Show That : i) If sn≥a for all but…

Q: Let a1 = 1 and an+1 = that {an} is increasing and bounded above. (Hint: Show that 3 is an upper…

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Q: b. The sequences Xt and Yt are such that for t ≥ 0 Xt+1 = 0.3Xt − 0.1Yt − 0.1 3Yt+1 = −Xt + 0.5Yt +…

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Q: Prove the following: If (xn) converges to x, if x 0, and if rn 0 for all n 2 1, then there Prove…

A: As per our guidelines we are supposed to answer only first question please repost other parts in the…

Q: Consider the recursively defined sequence s1 = 1 and sn+1 = 1 / (4 - sn) for n  ≥ 1. i. Prove that…

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Q: ? 4. Show that the sequence {fn}, where nx fn(x) = XER 1+n²x²⁹ is point-wise convergent but is not…

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Q: If P(x) is some polynomial then ΣP(n)x" has radius of convergence 1. n=1 Select one: O True O False

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Q: Consider the sequence defined by a, = In(2n³ + 2) – In(5n³+2n² + 4). Find the limit of an as n → ∞.…

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Q: Consider the recursively defined sequence s1 = 1 and sn+1 = 1 / (4 - sn) for n  ≥ 1. i. Prove that…

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Q: etermine and justify if ∞ In(x) k=2 (1+k) ³/2 converges or diverges.

A: given series ∑k=2∞ln(x)(1+k)3/2

Q: 1. Let x₁ = k> 0 and 2n+1 = √4xn-1 for all n E J. (a) Determine for what values of k the sequence…

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Q: Let {an}=1 be a sequence of real numbers that converges to A. Using only the definition of…

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Q: Let a be a nonzero real number. For any to satisfying 0 < x < 2/a, the recursive sequence defined by…

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Q: Consider the recursively defined sequence s1 = 1 and sn+1 = 1 / (4 - sn) for n  ≥ 1. i. Prove that…

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Q: Consider the recursively defined sequence s1 = 1 and sn+1 = 1 / (3-sn) for n  ≥ 1. i. Prove that sn…

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Q: Find the convergence set (or interval of convergence) for the 80 (x-2) Σ 6n+8 n=0 1.

A: Solution

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Q: Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is…

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Q: Prove that the sequence is monotone and bounded. Then find its limit. s1 = 5 and sn+1 = V4sn +1 for…

A: See the attachment

Q: Consider the recursively defined sequence s1 = 1 and sn+1 = 1 / (3-sn) for n  ≥1. i. Prove that sn…

A: Consider the given sequence:

Q: 1.2 Find the pointwise limit of the sequence of functions {fn(2)} = {nc(1 −z)} on [0, 1] and…

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Q: Define

A: Given sequence is: s_{n+1} = sqrt (5 + s_{n})  ................... (1)  where: s_{1} = sqrt 5 &gt; 0…

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Q: Let fn(x) x +n b. Use the Lemma for non-uniform convergence to show that the sequence does not…

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Q: Consider the sequence S n²r(1 – nx) x € [0, 1/n] - fn(x) := 0. elsewhere . Show it converges…

A: Let gx=n2x1-nx Differentiating wrt x, we get  g'x=n2-2n3x Putting g'x=0, we get  n2-2n3x=0⇒x=12n

Q: Let sn be a bounded sequence, and let tn converge to L. i. Show that if L = 0, then the product sn ⋅…

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