1. Let an = 5n 4n3 5 This sequence converges to L = 4 (a) Find the smallest N EN such that |an — L| < 10-4 for all n > N. (b) Show that, for all n > 12, |an - L| < 1/n. Deduce an integer M (not necessarily the smallest!) such that an - L| < 10-6 for all n > M. 1| < 10 101 a 5 (c) Prove from first principles that an converges to .

Q: True or false: Suppose {n} is a sequence of real numbers, and I is a real number such that for every…

A: Here the above mentioned statement is true.

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Q: PIUVE thlat Un 2 X9. A sequence {an} is defined recursively by ao = 1, a1 = 2, a2 = 24, and an = n 2…

A: The recursive relation has been proved in step 2.

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

Q: (c) Is the sequence {n-1}, a Cauchy sequence? Explain your answer. (d) Suppose that (a) is a…

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Q: 1 2 + xn (a) Prove by induction that în > 0 for all n € N. (b) Prove that (n) is a contractive…

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Q: strong induction ICt

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A: (a) Given that &lt;an&gt; is a sequence such that &lt;an&gt;→a, To Prove:  a1+a2+...+ann→a as n→∞.…

Q: Choose the correct answer. 1. The sequence (x) = (-n) is. a) Cauchy b) bounded e) divergent d)…

A: We have1) The sequence (xn)=(-n) is . . . . . . . . .Now we have take n=1 x1=-1put n=2x2=(-2) So, we…

Q: = 2√3 and bo = 3, and define two sequences recursively by √an.bn-1. Prove that {an} is monotonically…

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Q: 1. Prove that the sequence {4+} converges to 4.

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Q: Which of the following sequence is divergent? en {"} (-1)*+1 n s n² In+1 } {-3}

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Q: (2) Let an be the sequence defined by: a₁ = 1 a2 = 2 an = an-1 + 2an-2 for n ≥ 3.

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Q: (2) Let an be the sequence defined by: a₁ = 1 a2 = 2 an an-1 + 2an-2 for n ≥ 3. (a) Find a3,.. (b)…

A: Given that a1=1 , a2=2 and an=an-1+2an-2  for n≥3

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A: Solution; The arithmetic sequence is defined by Si=2+6i-1 We want to prove by induction that the sum…