Given f(n) € A(n), prove that f(n) E O(n²). Given f(n) E O(n) and g(n) E O(n²), prove that f(n)g(n) € O(n²³).

Q: [1] The forward-difference formula can be expressed as h h2 f'(xo) = [f (xo+ h) – f(xo)] f"(xo) f"…

A: Consider the provided question, The forward difference formula is given as,

Q: Given f(x,y, z) = In(x² + y² + z²), show that ² f , ²f , ² f x² + y2 + z2

A: Given problem:-

Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

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

A:

Q: diverges because an > 0. for all n 2 1. 2n + 1 True False

A: Given: The statement "an=n2n+1diverges because an&gt;0 for all n≥1." To determine: Whether the given…

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: Suppose E={O1,O2}, and F={O1,O3}F. If Pr[E]=0.6 and Pr[F]=0.61, wha

A: From the given information,   E={O1,O2}, and F={O1,O3}F. If Pr[E]=0.6 and Pr[F]=0.61   Here, the…

Q: Suppose that in the forward difference formula of order O(h) for f'(xo), h = 0.02 and |f"(c)| < 0.5,…

A:

Q: 1) Using k(x) = 2x + 1, m(x) = x² and find (k o m)(x) 2) Using f(x) = 2x + 3, g(x) = 3x + 5 and find…

A: Please refer the attached image for complete solution.

Q: at the superexpress checkout at the same time. Suppose the joint pmf of X, and X, is as given in the…

A: Solution.

Q: Let floor(1) + floor(1/2) + floor(1/3) + …· + floor(1/n) = ceiling(0/1) + ceiling(1/2) +…

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

A:

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: Is it true that if a, < b, for all nE N and E-1 a, is divergent, then E-1 bn is divergent? If false…

A: Given: an≤bn for all n∈ℕ To check that if ∑n=1∞an is divergent then ∑n=1∞bn is divergent

Q: Suppose that {$1,...Øn} F p and I F ø; for all i < n. Show that I F þ.

A: Proof: Suppose ϕ∈Γ∗ϕ∈Γ∗. As Γ∗Γ∗ is maximally consistent it clearly follows that there exists a…

Q: Prove that E, k2* = (n – 1)2"+1 +2 for all n > 1. =D1

A:

Q: Show that for any 1 ≤k≤n. T (1) = n²(n²-1²)...(n² (k − 1)²) 1.3.5 (2k-1)

A:

Q: If B' is of order 2x3 and BA is order 3x4, then A is of the order T

A:

Q: Show that for all z1, Z2 E C, Then we have: |Iz1| – |z2|| < |z1 +z2l

A: 2) The inequality can be proved as follows:

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

A:

Q: Prove e^z1/e^z2 = e^(z1-z2)

A:

Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

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: Suppose I is an open interval and f € C°(I). Show by induction that ffor х, а € 1, f(x) = f(a) +…

A: Given: I is an open interval and f∈C∞I   To show: fx=fa+∑k=1nfkak!x-ak+1n!∫axx-tnfn+1tdt for x,a∈I…

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: Show that E-o (-1)*(")=0, and then verify that () + C) + C)+ - = (") +C) + () +-- -G) + G) + G)-…

A: We can show given identity by definition of Binomial theorem:

Q: Define injections f : N → N and g : N → N by f(n) = 2n and g(n) = 2n. From f and g, the…

A: Given  f:N→N and g:N→N are two injective function defined by f(n)=2n and g(n)=2n Let 2n = k then…

Q: Show that for all n E Z, n² 5.

A: Consider the given condition: This shows that whatever will be the values of n, it will always be…

Q: 4. Prove C(n, r) = C(n- 1,r- 1) +C(n- 1, r) %3D using a combinatorial proof.

A: We know, C(n,r) = n!/((n-r)!r!)

Q: Prove that ( n2 + 1) / (n + 1) diverges to +∞

A: Let

Q: If x > 0, prove that T - In (1+ x) > 2+2x

A: Using derivative

Q: Prove that for all x > 0 and all positive integers n x2 e* > 1+x + 2! +3 +: 3! x" + n! Recall that…

A:

Q: Prove indirectly: "V n E Z, 4 ł (n² – 2)."|

A: Ans:Let us assume that 4/n2-2;∀n∈zthen we have there exist a positive integer k such…

Q: m Show that E (")C")= ("*"). Using Combinatorial Proof. \m+p. k=0

A: We will use equating coefficent method

Q: (iii) Prove that there is a constant C such that 0(2º) 1.

A:

Q: If X ∼ P oisson(λ), Prove that for any positive integer k, E[X(X − 1)(X − 2)· · ·(X − k)] = λ^ k+1

A:

Q: . Let X be a r.V. such that E(X") is finite for all n = 1, 2,... Use the expansion

A: Introduction: Suppose X is a random variable. Then, the moment generating function (mgf) of X is…

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

A:

Q: Show that f : N → R+ defined by f(n) = 3Vn+2 +n is O(n), but f(n) is not O(/n).

A: Big O notation is a mathematical notation that describes the limiting behavior of a function when…

Q: (12) If E(X*) is finite with E(X) = µ and Var(X) =o² then prove that E((X – µ)*) > of. %3D

A: Note : Since you have asked multiple questions, we will solve the first question for you. If you…

Q: If (an) has limit−1 and (bn) tends to infinity, does it follow that only finitely many terms of (an…

A: No it doesn't follow the statement

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, = 1, az = 2, b; = 0, b2 = 1, for n2 3, a, = 2an-1 + bn-1 for n22 b, = br-1+ an-1 express…

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Q: If det A= 0 why is det A+ = O? If A has rank r, why does A+ have rank r?

A: Given Data det A=0   Suppose the column and row of the matrix are linearly independent. The matrix…

Q: Prove that f(n) = 1000n5 + 20000n2 + 32 is O(n6 ); Is this a tight upper bound? Why or why not?

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Q: 2 O Expand Pn = 2x +3x +) in Term of P.f,.

A:

Q: Let f(n) = 2n – n? + 10n – 7. Show that f(n) is O(n³) using specific values of C and ng-

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Q: Suppose that in the forward difference formula of order O(h) for f'(xo), h = 0.2 and |f"(c)| < 0.5,…

A: hf''(x)/2 ≤0.2×0.5/2 ≤0.05

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.