Show that for all n E Z, n² 5.

Q: Let n E N*. a. Let a > 0. Then 1 as n 0o. b. - 0 as n → ∞ n16/5+2 c. (-1)"+1"" → 0 as n → 0 d.…

A: We know that limx→∞1x=0. It implies that if x→∞, then 1x→0. We need to calculate the limit for every…

Q: For any n e N, let An = {x € R | –n <x < n}. Prove that NneN An = A1. (Recall that we define 0 ¢ N.)

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Q: Show in detail that E [(X-µ)^n] = 0 for ∀n , where X~N (µ, σ²) and n≥3.

A: It is an important part of statistics. It is widely used.

Q: and g(n) is O(H), the

A: Given: f(n)=O(n2) and g(n)=O(n) We have to check that given statement is true or false.

Q: If (ar + br) = 2n %3! n+1 and k-1 8. E(ak - br) = n+2 - , then ar %3D k-1 k-1

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Q: 3. Prove the Archimedean Property, if x ER. then there exists K E N such that r SK.

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Q: (b) Define fo = 0, f₁ = 1 and fn = fn-1 + fn-2 for n ≥2. Show that 3 fak for all k ≥ 0.

A: As per the guidelines we shall solve one question only as you have asked more than one different…

Q: Let a, b eR with a < b. Assume that a + <b - for all n E N. Show that b- = (a, b). %3D n n=1

A: We will show  two sets are equal.

Q: Q3: Prove that (9":n e Z) (3":ne Z) but (9":n E Z) (3":n E Z).

A: We have to prove that 9n:n∈ℤ⊆3n:n∈ℤ but 9n:n∈ℤ≠3n:n∈ℤ

Q: Q5 Show that EL >². n n =1

A: Consider that x is a random variable. From the formula of population variance;

Q: Show that when |+ 1|<1, = =1+E(n+1)(=+1)"

A:

Q: (a) Find the least positive k E N such that 3* =10 1. (b) Using this k, prove: for all m e N, 3m =10…

A: (a) find the least positive k∈ℕ such that 3k≡10 1 b) using this k prove that : for all m∈ℕ 3m≡103m÷k…

Q: Q3: Prove that (9":n e Z} S {3":n Ee Z} but {9":n e Z} (3":n E Z}.

A: To prove:  {9n:n∈ℤ} ⊆{3n:n∈ℤ} but {9n:n∈ℤ} ≠{3n:n∈ℤ}

Q: Prove that L ={(1Z*)"10" | n e N}is not regular.

A: The given language: Then, the above language can be rewritten as, To prove that L is not regular.…

Q: Let a, b eR with a < b. Assume that a + <b - for all n e N. Show that n n=1

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Q: Show that for any n E N, 7(15".

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Q: 3. For all æ E Z[i], there are N (x) elements in Z[i]z.

A: The set of Gaussian integers is a subset of the set of complex numbers containing the elements with…

Q: What is the general rule to evaluate I N,where N is any positive

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Q: 2. Prove that, for all k E N, 2 can be written as the sum of two squares.

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Q: If E(0) = 0 then is said to be a(n)

A: if Eθ^=θ then θ^ is said to be unbiased estimate for θ.

Q: For all n E N, let An = {0, 2n, 4n, 6n, . . {x € Z|(3n E NU{0})(x = 2n)}. (a) Find (with proof) [U…

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Q: For any positive interger n, (n + 1)! / n! - n = (A) 0 (B) 1 (C) n (D) n + 1 (E) n!

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Q: Show that E [(X-µ)^n]=0 for ∀n, with X~N(µ, σ²) and n≥3 being odd numbers.

A: We need to show; E[(X-μ)n]=0 for ∀n and n≥3 being odd numberswhere;X~Normal(μ,σ2) That is, we need…

Q: Show that for any n E N, n(n + 1)(2n + 1) k=0

A: The solution is given as

Q: Prove that for each n E N, (d) (n) e (d) dn

A: To prove, for each n∈ℕ, nφn=∑d|nμ2dφd. We know that the mobius function μn is defined as: μn=1if…

Q: that for n > k we Sn+1 ||

A: The expression given is ,  n+1k+1=∑j=knnn-jjk The number of ways to partition n+1 objects into k+1…

Q: Show that for all n e Z, n? – 3n +17 2. -

A: Given: For all n∈ℤ, n2-3n+1≠2

Q: Prove the following: 1) 1+3+5...+ (2n-1) = n²

A: *By Bartleby policy I have to solve only first one as these are all unrelated  problems.These are…

Q: (5) Show that T(n)o(n) > n for all n E N. 11

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Q: Prove for r ≠ 1 that ∑_(k=0)^(n-1)(rk)=(1-rn)/(1-r). If ∑_(k=0)^(n-1)(rk), then compute s(1−r) =…

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Q: Let n be a positive integer, then n+1≥ for all n.T/F

A: We need to show that n+1≥n for all n where n is a positive integer. Lets proof by contradiction

Q: Show that E(r(r + 1) =- n(n + 1)(n + 2), n E N.

A: We have to show that ∑r=1n(r(r+1))=13n(n+1)(n+2)           ∀n∈ℕ

Q: O N (AT) = R(A)-

A:

Q: 15. prove that L. d= nd(n)/2

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Q: Suppose E(X)=2, E(Y)=4, E(Z)=10. X, Y, Z are dependent. What is E(2X+3Y+Z)? Can not compute with…

A: Given,E(X)=2E(Y)=4E(Z)=10X,Y ,Z are dependent

Q: Use Gauss's trick to determine H(n)=1+5+9+13+...(4n-7)+(4n-3)+(4n+1) I got n(4n+2)/2 is this…

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Q: 1. For an = 5n2 + 9n3 + 1, an = n2, prove or disprove: rn = O(an) (n → o)?

A: We will use the standard definition.

Q: es lls ew zeiflucifiib ons einiseen stigesber Show that for neN, (2n)! ' (r!)² ((n −r)!)² - 7=0 2n -…

A: We will use binomial expansion to prove given relation

Q: For xn = 5n² + 9n³ + 1, an = en = n², prove or disprove: an = O(an) (n → 0)?

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Q: 4|5" – 1, Vn e N

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Q: Show that for any n E N, n(n + 1)(2n + 1) %3D k=0

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Q: For any n e N, let Bn = (n, ∞0). {B„}1 is descending. Show that Bn = 4. %3D n=1

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Q: Let a, b e R with a < b. Assume that a + <b - for all n E N. Show that 1 = (a, b). a + ,b n n=1

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Q: If var(e) → 0 as n → 0, then 8 is said to be

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Q: (b) Z/n = (q) if and only if kq 0 for all 1 < k <n.

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Q: (-1) is where n is an integer. equal to the above e(1+2n)i where n is an integer. equal to the abov…

A: We know from eular formula e^iθ = cos θ + isin θ

Q: Suppose that X ~ Unif(-1,1). Compute (d) P(X = 0.5), (e) P(X² > 0.5).

A: Let ,  The pdf of X is ,  = 0 ; otherwise Our aim is to find the following :  d) P(X=0.5) e)…

Q: Find all m, n E N such that: n! + 2n + 49 = m²

A: This problem can be solved by graphical approach. Use a graphing utility to plot the graph of…

Q: 4. Prove that n(3n – 1)r x + 4x + 7x + ... + (3п - 2)x for n e N.

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Q: Show that yn Z, if n³+5 is odd then n is even

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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.