Theorem 5. Abel's Lemma, or Partial Summation Given Sn = a1 + a2 + + an ... (a) a1v1+...+ a„Vn = $1(v1 – v2) + ...+Sn–1(Vn–1– Vn)+SnVn (b) If m < a1 + .. + an < M Vn, and vn is positive and decreasing, then mvị < a¡v1 + ... + anVn < Mv1. + ɑnVn < Mv1.

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A: We need to prove that, if f and g are functions, then f ∪ g is a function.

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A: As required in the question, detailed proof of 3, 5 and 9 are given as follows :

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Q: 12. Prove that if gcd(a, n) { b, then ax = b (mod n) has no solutions.

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Q: a) and (b) below. Use a 0.01 significance level for are from populations with the same mean. heses?

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Q: (c) If in (b) |sn < M Vn, then |a1v1 + ... + a„Vn| < Mv1 Vn.

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Q: 8) Suppose A E Mnxn(F) is diagonalizable. Let A1,· , Ak be the (distinct) eigenvalues of A. a) Prove…

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Q: Use the reason for the proving are theorem 113,114,115,116 and 117

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Q: 29. For any odd integer n, n° n² +3 %3D 4 4

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Q: H 28. For all integers a, b, and c, if a b and a/c, then al(b + c).

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Q: Expand the expression for Mr Taylor.

A: The previous solution is incorrect. Please follow this one.

Q: Problem 2. Suppose that A is similar to diagonal matrices D1 and D2. Show that every eigenvalue of A…

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Q: The data show the chest size and weight of several bears. Find the regression equation, letting…

A: chest size( inches) (x) weights (pounds) (y) 58 414 50 312 65 499 59 450 59 456 48 260

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Q: 8. Let V and W be the vector spaces and T : V → W be linear. (a) Prove that T is one-to-one if and…

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Q: 11. Prove that if f is continuous on (a, b] and lim,a* f(x) exists, then fis uniformly continuous on…

A: The theorem is done .

Q: State and prove Littlewood's Third Principle

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Q: 1. Solve the equations below. Show your work. a.) x2 – 14x + 45 = 0 b.) 3x2 – 2x - 5 = 0

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