6. Let fn (n = 1, 2, 3, ...) be respectively: (a) n² X[1 1 (b) (c) X[n] 1 3kX[k,k+1] n k=0

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A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

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Q: 28. Show {Xn, n ≥ 1} are independent if o(X₁,..., Xn-1) || σ(Xn) are independent for each n ≥ 2.

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Q: 1. Let f(x) = x² + 1. %3D (a) Compute f(0), f(-1), f(1/2), and f(/2). (b) For what values of x is it…

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Q: #8: Suppose that an = as 8n a₁²2² +2a₂x³ + 3a3x²4 + 4a4x³ + ... (i) Σf(n)x"+2 n=0 (ii) g(n)x+1 We…

A: This is a good exercise in summation and we shall see how to change the indices.

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Q: 8. Find an injective function from P(N) × P(N) into P(N).

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Q: L. nft 7r 2x+1 1 4) f(x) = 2 I- 3 < 1 f(1)= f(-2)= f(6)=

A: The given expression is:

Q: 1. Let N = {1, 2, 3,...} and let f: N→ N be a function given by f(x) = 2x + 3. (i) Find f(f(4) + 1).…

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Q: Let Calculate f(x) = ² Σ K=O k f(K) 2 k! where K=0,1₁.. X>0

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Q: 2. Let f: RR by f(x) = -2x + 1. Let A = [2,5] B = [-1,3] C = [−2,3] D = [1, 4]. Find each of the…

A: Here are the solutions:  Note:We have provided the best possible solution for you. Kindly, verify…

Q: 1.) Let f(x) = 7 + What is f(3 + h)?

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Q: 3. Let r = (4, –3, 1, –5) € R'. Which of the following equals ||r||? (a) 5 (b) 13 (c) V51 (d) v63

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Q: 6. Show that (x, y) = 1/2 (||x + y||² − ||t||² — ||ý||²) -

A: Given; &lt;x,y&gt;=(llx+yll2-llxll2-llyll2)/2.

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Q: 14. Let f(n) = n² + 20 and g(n) = n³. Show that f(n) is O(g(n)) by finding a positive integer N for…

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Q: 1. Let k and n be non-negative integers. (a) What does the function pr(n) count? (b) Show that the…

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Q: determine L−1[F].

A: