Prove that for any integer a, [a]=m = [r]=m where r is the unique remainder when a is divided by m, and thus, A/ =m = {[0]=m: [1]=m ..,[m – 1]=m} =: Zm. %3D

Q: Find the Taylor polynomial of order 3 generated by fat a. f(x) = In (x + 1), a = 7

A: The standard form of Taylor polynomial of degree 3 at a is given as: f(x)=∑n=0∞f(n)(a)n!x-an…

Q: Find the principal ideal (z) of Z such that each of the following sums as is equal to (z).   (4) +…

A: Known fact: m+n=d    where d=gcdm, n

Q: Let X Gamma(a, B). Prove that for any positive integer n > 1, E(X") = [(a +n– 1)(a+n – 2)..-a]3".

A: It is given in the question that X~Gammaα,β then we have to prove that for any positive integer n≥1.…

Q: 1 Find the characteristic polynomial Ac(t) for the matrix C and evaluate 3 1 Ac(C) to verify that…

A: The characteristic polynomial is obtained by solving |C-tI| where C is the given matrix t is the…

Q: 2. Consider the following set of linear polynomials with rational coefficients: E = {ar+bla.be Q. a…

A: The set E1 is defined as follows. E1=ax+b | a, b∈ℚ, a≠0 The set is linearly polynomial with rational…

Q: Let p be a prime and let q = p². Show that Z,[x] has a unit that is not a constant polynomial.

A: Given that, Let p be a prime and let q = p ^ 2 . Show that Zq[x] has a unit that is not a constant…

Q: For any n > 1, prove that the irreducible factorization over Z ofxn-1 + xn-2 + . . . +x +1 is π…

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

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Q: Show that the expansion of the function f(x) = a2r+1 %3D in a series of Hermite polynomials of odd…

A: Solution :-

Q: The number of irreducible polynomials over Z13 of the form x + ajr + a2 is 91 O 182 O 78 O 169

A: Given x2+a1x+a2 is a 2nd degree polynomial .For a 2nd degree polymonial , no of irraducible…

Q: For any k e Z, give a formula for the coefficient of x* in the expansion of (2-!)" 100 x²

A: Given expansion: x2-1x100 The given problem can be compared to the general form x+an. Here, n=100…

Q: Which of the following is homogenous of smallest degree? (A) In x - In y B x+y x-y C √x-y x² + y4 X…

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Q: Show that a fraction r = a/b in lowest terms has a finite decimal expansion if and only if…

A: Given, r = a/b, where b = 2n5m for some n, m ≥ 0. We know, that any integer divided by 10N has a…

Q: Show that 2 is equal to the product of a unit and the square of an irreducible in Z[i].

A: We have to show that 2 is equal to the product of a unit and the square of an irreducible in Zi. The…

Q: The number of polynomials of degree 3 in Z5[x] is: O (4^4)*5 O 5^4 O 4^4 O 4*(5^3)

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Q: e of all polynomials in C[0, ed in C[0, 1] with respect to

A: To prove a subset W is a subspace of V, it must satisfy 3 conditions: 1) 0 ∊ W. 2) closed under…

Q: Let a be a non-zero element of Z43. Explain what it means for |a] to be of order w, and hence prove…

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Q: Find the Taylor expansion of f(x) = (x2 + 7)(7/3) for x near to 1, up to and including the x2 term.

A: Taylor series

Q: Use the third Taylor polynomial P3 (x) for f(x) = x² + Inæ about o = O 1 000 1 and evaluate f(2).

A: Given that fx=x2+lnx. We have to evaluate f2 by using the third Taylor polynomial P3x for fx=x2+lnx…

Q: 3. Find the fifth Taylor Polynomial for f(x) = In x around 1.

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Q: (b) Prove that for any positive integer n, Ej. 2 = (n – 1)2"+1 + 2 (c) n(n² – 1) Prove that for any…

A: b) Let the statement be P(n): 1·21+2·22+3·23+⋯ +n·2n(, or,∑j=1nj·2j )=(n-1)2n+1+2. For n=1, P(1) :…

Q: 9. Given the Mac expansion of y = In(1+ x) below, use it to approximate In(0.4) to the 4th decimal…

A: We will use given formula.

Q: 1. Prove that if A is infinite and a e A, then A – {a} is also - infinite.

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Q: Show that for the Legendre polynomials Pn the following holds for all n E N (n+1) Pn+1(x) − (2n…

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Q: Let p be a prime divisor of the Fermat number Fn 22" + 1. : 2n+1 (a) Show that ord, 2 (b) Use…

A: As per the question we have the Fermat number Fn = 22ⁿ+1 and its prime divisor p. Here we will use…

Q: Attached is the question I'm needing help with. Thanks in advance!

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Q: Prove the following for Legendre polynomial P(x): 0 for m # n P,(x)P(x)dx 2 for m = n 2n+1

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Q: Q2/ Determine the positive real root of In(x³) = 0.7 (a) using three iterations of the bisection…

A: There are different methods to find the real root of a function using initial guess. Two such…

Q: Find the Taylor Polynomial of degree 3 and estimate the Remainder of in X0 = 0 for x = 0.1

A: Consider the given function whose taylor polynomial is required to calculate.

Q: S Prove by mathematical induction that x 1 is a factor of x" 1 for all integers n > 1.

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Q: Find all distinct principal ideals of Zn for the given value of n. n = 7

A: Find all distinct principal ideals of Zn for  n=7.

Q: Write out the complete proof of the integral remainder formula in Taylor's theorem.

A: proof of the integral remainder formula in Taylor's theorem.

Q: Let f (x) = 1/1+5x The coefficient of x6 in the 6th Taylor polynomial p6 is  . (Write your answer as…

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Q: 5. Let a, b, N E Z where N > 0. Prove that [ab] = [[a][b], where [r] denotes the remainder of r…

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Q: The first-degree Taylor polynomial of e" is 1 – x. Select one: O True False

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Q: prove that and = nd(n)/2

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Q: Let f (x) = e4x. The coefficient of x6 in the 6th Taylor polynomial p6 is

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Q: 1. Prove by mathematical induction that for all positive integral values of n, 2=1 =- (n + 1)(2n +…

A: (1)-According to question given that∑r=1n r2    ....(1)Now  apply the sum formula we get that∑k=1n…

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

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Q: snip

A: Given, A=1-123

Q: 1 Let f (x) 1 + 3x The coefficient of x8 in the 8th Taylor polynomial pg is . (Write your answer as…

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Q: Let f (x) = e3x. The coefficient of x in the 5th Taylor polynomial p5 is

A:

Q: Use Taylor's Theorem with n = 2 to approximate 1+x, x > –1. Provide an expression for the remainder.…

A: The Taylor series expansion with n=2 is as follows:…

Q: Prove that for any integer a, [a]=m = [r]=m where r is the unique remainder when a is divided by m,…

A:

Q: Prove that for any positive integer n, Vn is either an integer or irrational.

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Q: 1. Find the least no EN such that 10k + 3 no, it has the stated property. and prove that

A: Given: 10k+3<2k, ∀k≥n0, where n0∈ℕ is a least one. To find: Value of n0. To prove: 10k+3<2k,…

Q: Let a E Z, a #0. Prove that for any positive integers m, n with n<m, a2" +1 divides a2 -1. a) a even…

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Q: Use the second Taylor polynomial P2 (r) for f(z) = r? + In(1 + z) about zo = 0 and evaluate S…

A: we have given f(x) = x2+ln(1+x) we have to find second Taylor polynomial for f(x) about x0 and then…

Q: Let f (x) = cos(x). The coefficient of x6 in the 6th Taylor polynomial p6 is  . The coefficient of…

A: Taylor series expansion of f(x)=cosx about x=0

Q: Take non-zero positive integers m and n. Show that Z/mZx Z/nZZ/mnZ if and only if m and n are…

A: Given That:To take non-zero integers m and n .   ℤmℤ × ℤnℤ ≅ ℤmnℤ if m and n are relatively prime ,…