Let o,T E Sn for some positive integer n satisfy o(k) 1< k < n. Prove that o-(k) = k and p(k) = k where p= To. k and t(k) = k for some k with

Q: 3.3 Prove that E does not converge uniformly k! k=0 on R.

A:

Q: If the function (x) =sin 2(2n+1)x for some positive integer number n, then the conclusion of Rolle's…

A: Given that: f(x)=sin2(2n+1)x on the interval π8, 3π8.and satisfies the Rolle's theorem at…

Q: Prove that if h E L'(R) and [ h= 0 for all s E R, then h = 0.

A: Use the Leibniz integral rule: For ∫axbx fx, tdt Where -∞&lt;ax, bx&lt;∞ We have, ddx∫axbx fx,…

Q: Let p». Prove that, for every e > 0, a.a.s. the largest component of G(n, p) has size at least…

A:

Q: Prove that the set S = {n² : n e N} is denumerable. %3D

A: We have to show that the given set S is a denumerable set.

Q: Prove using the IVT. For all positive integers k, cos x = xk has a solution.

A: Intermediate value theorem: Let f be a function continuous on a closed interval a,b. Let k be number…

Q: Consider the set A = {1/2^{n-1}: n\epsilon \mathbb{N}}. Show that Inf(A) = 0. WITHOUT USING LIMITS

A:

Q: Find the value of k such that A is singular. 1 0 3 4 -1 0 2 4 k k =

A:

Q: Prove that Hence deduce that L{x"; p} r(n + 1) pn+1 " if n>-1, p > 0. L{x-¹/2; p} = √√(π/p),p> 0

A:

Q: The Euler totient function is defined as ϕ (m) = |{k : 1 ≤ k ≤ m gcd(k, m) = 1}|,  or rather the…

A: Introduction: If p is a prime number and p|n, then ϕ(pn)=pϕ(n). Proof. You'll see that every number…

Q: Prove that gcd(m + 1, n + 1)|mn - 1 for all m, n E Z \ {-1}.

A: Let, gcd(m+1, n+1)=x From the properties of GCD, we can say that, x divides m+1 and n+1 Now,…

Q: Σ does not converge uniformly on R. k! k=0 Prove that

A: We know that ∑k=0∞xkk!=ex. So, we will show that ex does not converge uniformly on ℝ.  Let us assume…

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: does not converge uniformly k! k=0 Prove that on R.

A:

Q: Show that F0F1 . . . Fn−1 = Fn − 2 for all n ≥ 1, where Fi is the i-th fermat number

A: To Show -

Q: Prove or disprove that 0 ≤ x2 - xy + y2 for all x and y in Z.

A: Consider the given inequality 0≤x2-xy+y2  for all x and y in  ℤ Now,…

Q: For allne z, gcdin,nt)=l.

A:

Q: Prove by induction that 1 2. Prove that (xn) is contractive and thus convergent. Compute lim (xn).

A:

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: Suppose that 0< fn < In on X for all n e N. If gn 30 on X, then fn 30 on X.

A: Solution:-

Q: For any positive integer m and any interval J C [0, 1] satisfying |J| 1 is uniformly distributed in…

A:

Q: Let Sn=Vn. Find an and show that an O but that Sn=ER=1@k diverges. ->

A: Solution:-

Q: Prove that Theorem 13.1 (vi) using the simple ftn, nonnegative ftn and general ftn. (vi) If X is…

A: vi) From the given information, X is measurable in N.  There exists a sequence of simple functions,…

Q: The Euler totient function is defined as ϕ (m) = |{k : 1 ≤ k ≤ m gcd(k, m) = 1}|, or rather the…

A:

Q: 3. Let E be a bounded subset of R such that E = U₁1 Ej. Prove that if the E, are of Lebesgue measure…

A: Lebesgue Measure : A set E⊂ℝ is said to be Lebesgue measurable if for every subset of A of ℝ…

Q: Prove the following in Z[i]: if a y and 3|y with a and B relatively prime then aß | y. in

A:

Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

Q: Find the smallest integer in the given set a.{ xϵ Z f x &gt; 0 and x = 4s + 6t for some s, t in Z }

A: Consider the equation: x=4s+6t……1   where: x, s, t∈ℤ and x&gt;0

Q: In the domain of Gaussian integers Z[i], the element 15 is 青 O Reducible O Unit O Irreducible O None…

A: Since 3∈ Z[i] and 5∈ Z[i] And 15 = 3 × 5 Hence 15 is reducible element in Z[i]

Q: For any k = 1, 2, .... n, find E T₁+T2+ T₁+T2+ ... + Tk + In,

A:

Q: Let m and n be positive integers with gcd(m,n) = 1. Prove that Z/mZ x _Z/ nZ = Z/mnZ

A: Suppose m and n are relatively prime.

Q: Compute the convolution x[n]*h[n] for all n> 0, by using Graphical or Tabular method. x[0] = 3, x[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: In the proof that |(0, 1)| > |N|, we use Cantor's Diagonal Method, where we change the nth digit dnn…

A: We are given the following. In the proof that |(0, 1)| &gt; |N|, we use Cantor's Diagonal Method,…

Q: If m, n are relatively prime, then o(mn) = o(m)o(n)

A:

Q: Question attached.  I have found solutions to this probem on the internet, but I don't understand…

A: Let N be a given positive integer and let p be the least prime number greater than N + 1.Let n be an…

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: 5b) Let f (x, y(x)) = y(x) + 1 and y(0) = yo = 2. Use Picard-iteration and induction to show that YN…

A: Please go through the solution.

Q: - In the domain of Gaussian integers Z[i], the element 43 is O irreducible O reducible O unit

A:

Q: Let a > 1 be an integer. Prove that n|p(a" – 1), where o is the Euler ç-function.

A: In number theory, Euler's totient function counts the positive integers up to a given integer n that…

Q: Let x, y, z ∈  ℕ. Suppose gcd(x, y) = 1. Prove that if x | yz, then x | z

A: In the given question we have to prove that if x|yz , then x|z and g.c.d(x,y)=1 and x,y,z∈ℕ. we will…

Q: Q1. Prove that KT = = -.

A:

Q: Prove that : a"+ (a+ 1)° =1 +a(1+い) (mod a la+s) for all a, m,n E Z.

A:

Q: Find the value of k such that A is singular. 0 4 6 -1 0 4 2 k k =

A:

Q: Prove that the sequence 1 f(1n) j(zn) 2 f(In) F(x,) ' f(r,) 1 with Xn+1 = n an = %3D an is cubically…

A: Sol

Q: Find infinitely many polynomials f(x) in Z3[x] such that f(a) = 0 forall a in Z3.

A:

Q: Prove that for all k E N, k ose [coske -sink0j cose Lsinke cosko

A:

Q: Prove that for any k > 0, nk e o(2v™).

A: The complete solutions are given below

Q: Prove that det(kA) = kn det(A) for any A ∈ Mn×n(F).

A: In determinant the scalar quantity is taken common from a single row or a column.