1.16. Prove that the Stirling numbers satisfy the following relations: n (c) Ĉ (−1)ª¯¹ (i − 1)!S =0,_n>1, - i=1

Q: 10. Let w = f(z) = Find f(1 - i) 1+z 1-z A. 1 - 2i B. −1+2i C. -1-2i D. none of the above

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Q: If f (x) = x² – 4, what is the simplest form of +1), %3D - X -1 Select one: a. х + 3 о ь. х — 1 c. x…

A: The given function is: f(x)=x2-4

Q: Suppose that a new company begins its operations in 2020 and the size of the company over the next…

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Q: 7. Let A = fa; b; c; dg and B = fx; yg. (a) How many functions are there from A to B? How many of…

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Q: 6. Prove that for x > -1 then Vn E N, (1 + x)" >1+ nx .

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Q: The randon variable x has dencity že" Find P(1xl +1X-31< 4).

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Q: What is the domain of the profit function P= 11w+ 6z? O A. Defined for all w 2 0 and z> 0 B. Defined…

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Q: p(x) = a + ax + a,x² + ... + 10x?

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Q: (i) Construct the minimax polynomial p2 E P2 on the interval (-1,1] for the function g defined by…

A: As per the question we have to find the bset fit polynomial in p2 ∈ P , to to the curve g(x)=sin(x)…

Q: 5. (a) Verify that t(n) = t(n+ 1) = t(n + 2) = t(n + 3) holds for n = 3655 and 4503. (b) When n =…

A: To find-  Verify that τn = τn+1 = τn+2 = τn+3 holds for n = 3655 and 4503 When n = 14, 206, and…

Q: 5. Suppose that f(n) satisfies the equation 5. f(m + n) = f(m) + f(n) – 2f(mn) for all integer m,n…

A: The objective of the question is computing the value of f(n).

Q: 14)- Consider the following formula f"(x,) = a-4fo+3f-1 _ 2hfm(C) 6h2 3 Find the optimal h.

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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.

A: Given::- Show (Xn, n ≥ 1} are independent if o (X1,..., Xn-1) || σ (Xn) are independent for each n ≥…

Q: 12. (a) For n = 3, explain why Lo(x)+ L1(x) + L2(x)+L3(x) = 1 for all x. Hint: It is unnecessary to…

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Q: Suppose a and b are positive integers and a>b. Determine the number of pairs (a, b) that satisfy the…

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Q: Prove that (a) P(En F) = P(E)- P(En F) (b) P(EnF) =1-P(E) – P(F) + P(En F)

A: Solution: 6. Here, it is needed to prove that

Q: 63. Given n nonzero numbers o1,...,on, show that the minimum value of S(x1....,xn) = xfo7 +..+ x…

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Q: 5. Given n constants ki (with i = 1, 2, ..., n) and n functions f(x), deduce from Rules Iv and V…

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Q: 6. Let f(x) = 20/(x6 + x4 + x² + 1). %3D (a) Show that for any k e [5,20], there is a point c e…

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Q: 35. Prove: If xn = :o(an), then xn = = O(an). Show that the converse is not true.

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Q: Let P(x) be a polynomial that interpolates f(x) at x; for i = 0, 1,...,l−1,1+1, ---, n (i.e.,…

A: Given Information:The polynomial  is a polynomial which interpolates  at ,  exclude .The polynomial…

Q: Define m t(m) = Σ(2n – 1). n=1 For a given m this is the sum of the squares of the odd integers up…

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Q: Let n EZ. Show that f(x)= x³ – nx +2 is irreducible over Q for n+-1,3,5. 72

A: Given: fx=x3-nx+2, n∈ℤ. To show fx is irreducible over ℚ for n≠-1,3,5.

Q: 5. What is the number of surjections f: [9] → [6]? Use this information to determine the Stirling…

A: We have to find the number of surjections, f9→6 Where,, 9=1,2,3,4,5,6,7,8,96=1,2,3,4,5,6

Q: Show that for f(n) = n³ - (ri'log, n) + 2n E O(n³ ) A give the constants: (c, n) B:arguing that your…

A: When a function is bounded above by another function with a factor asymptotically, it is referred to…

Q: Let H(x):=k for x=, (keN) and H(x):=0 elsewhere on [0,1] Show that H(x) 4ER[0,1].

A: In this question, we use the definition of Riemann integral to check the function H(x) is Riemann…

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: 2. Let f(x) = -x– 1. a) Find f-1(x).

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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.

Q: A := {x € (0,1)|x= ar, for some r € Q} re that sup(A) = 1.

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Q: 5. Let h (z) = (z+1)(z+6)(z+10) z(z+2)(z+3) (z+3)(z–2)°, (z+2)(z+6) What is the residue of h (z)at z…

A: Given a function on complex plane as hz=z+1z+6z+10zz+2z+3+z+3z-22z+2z+6 Residue at z=-2 is given by…