Theorem 1: Suppose f(m, n) is a double sequence on R such that f(m, n) > 0. Then, E f(m, n) converges if and only if the set of partial sum is bounded. Proof. The detailed is left as exercise. Just only apply monotone convergence theorem. Theorem 2: If E f(m,n) converges absolutely, then Ef(m,n) converges. Proof. Left as exercise.

Q: Theorem 2. Suppose u e F where F is an ordered field. Then u is positive f and only if u > 0.…

A: Note: According to bartleby we have to answer only first question please upload the question…

Q: Which of the following is the derivative of f (g(h(x)))?

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Q: Theorem 23. If u, v E Q then we can find integers a, b, d with d > 0 such that u = a and v = d'…

A: By Division Algorithm we know that,               Let a, b be two integers with b>0. Then there…

Q: 4. Give an example of an indefinite quadratic form on R³ whose corresponding symmetric matrix has no…

A: Let Q(x1,x2,x3)=-x12-2x1x2+6x1x2+3x22-2x2x3-x32 be a quadratic form whose symmetric matrix is…

Q: Prove that if D is an integral domain with unity that is not a field, then D [x] is not a Euclidean…

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Q: Prove Theorem

A: (a)   Let A=a11a12...a1na21a22...a2n⋮⋮⋱⋮am1am2...amn…

Q: IfX is a binomial distribution variable, then, the moment generating function is (pe' +q)" .

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Q: Prove property 4

A: Let ϕ is an isomorphism from a group G onto a group G. We have to prove ϕK=ϕk|k∈K is a subgroup of…

Q: Let a, b, and e be integers, where a # 0. Then (i) if alb and alc, then al(b + c); (ii) if alb, then…

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Q: Prove this theorem.

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Q: Theorem 31. Let F be an ordered field with ordered subfield Q. Then F is Archimedean if and only if…

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Q: A survey of 12,000 adults and children in the United States has shown that many people have lost a…

A: Given, A survey of 12,000 adults and children in the United States has shown that many people have…

Q: (b) If m < a1 + ... + an < M n, and v, is positive and + anVn < Mv1. + anUn| < Mvi Vn. decreasing,…

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Q: C) Give a convincing argument that [ f(x)dx=1 Hint: Show that 01, and for b>1, Se-x/² dx →0 as b→∞.

A: To fully understand and work with the problem and the argument provided, you should have a strong…

Q: ry lim (z.v)¬(0,0) + y² exists.

A: Please refer the attached image for complete solution.

Q: Q3: Prove that i) Intersection is distributive over union for arbitrary number of sets. ii) Union is…

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Q: 10,000 1) Suppose a population is modeled by P(t) What are the initial population and the 1+199e-…

A: The given model is: P(t)=10,0001+199e-0.1t...(1) Where, t is time and P is the population.

Q: Theorem 4. Suppose x, y, z e F where F is an ordered field. If x < y then x+ z < y + z. Exercise 5.…

A: From the given information. The variables x, y, z belong to an ordered field F such that x&lt;y.

Q: Theorem 6.1.2 (Differentiation of the Integral) If f is continuous on [a, b), then F(x) = S f(t) dt…

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Q: Exercise 36. Prove the above theorem. Hint: some pieces have been done in earlier theorems and…

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Q: Theorem 14. Properties of the Gradient Let f and g both be real-valued functions of two or three…

A: Given that functions f and g are differentiable.

Q: Prove Theorem 3.6.

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Q: Theorem 8.3.3 If X ~ t(k), then E(X2r + 1) = 0 and µz, = k"- (2r – 1)(2r – 3) ... .1_ (k– 2)(k – 4).…

A: Answer: For the given theorem, The proof is:

Q: Describe in words and sketch the following surface, in the 3-dimensional coordinate system. Use the…

A: Given: The equation is x2+y2+z2-6y+2z=4. To draw: The graph of the given equation.

Q: Prove Theorem 21.2 and Theorem 21.3.

A: note : Since you have posted multiple questions, we will provide the solution only to the first…

Q: Theorem 43. Suppose (a;);2no is a constant sequence with value a. In other words, suppose a; = a for…

A: We are given a sequence aii≥n0 and ai=a ∀ i i.e., it is a constant sequence. We can write ai=a1, a2,…

Q: 7.2.9 Additivity Theorem Let f := [a,b] → R and let ce (a, b). Then f E R[a, b] if and only if its…

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Q: Prove this theorem.

A: First Isomorphism theorem for Rings: Let ϕ be a ring homomorphism from R to S. Then the mapping…

Q: Theorem 7. Every element of R has an additive inverse. More specificially, if x = [(a,)], then -æ =…

A: The solution is given as follows

Q: 3i16 - 103 1+i 2i84 + i79

A: given the expression 3i16-i1032i84+i79 we know that i2=-1…