2. Show that for all n ≥ k ≥ 0, n (1)-(²) = k k

Q: Let a, b be real numbers such that la], [b] >k> 0. Show that la - bl k² a ≤

A:

Q: 17. The interval for function Is+3|2 1/2 is (-7/2,- 5/ 2] 0(-7/2,-5/ 2) (-00,-7/2)u(00,-5/2)…

A: Given: s+3≥12 We have to find the interval in which the given inequality satisfies.

Q: 1' 1 3 (e) Show that 5 k=1 k k+2

A: I am attaching image so that you understand each and every step.

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

A:

Q: The number of visitors P to a website in a given week over a 1-year period is given by…

A: Given data :  P(t) = 117 + (t - 88)e0.02t 1 < t < 52 P'(t) = d/dt (117 + (t-88)e0.02t…

Q: 2. Show that 7-(4-3 k=0 (-1) k + 1 \k n+1

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Q: (b) Determine the value of k for which p(x) = k (0.01)*, x=1, 2, 3, ... x-1

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Q: 10) Define the metric d on R² by d (x₁, y₁), (x2, y2)) = max{|x₁ - x₂], ly₁ - yal}. Verify that this…

A: This is a question of topology.

Q: Let S(2) = ) for |z| <1. 1 n=1 zn Evaluate f(10) (0). You may leave factorials in your answer…

A: In the question we are given a complex function f(z) in terms of infinite series, and we have to…

Q: 5. Find the domain off(z-In(e*-3). Then find f-1 and its domain.

A: f(x) = ln(ex-3)The function "ln" is defined for positive numbers only.Hence, x should be such that…

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: k It is known that ( + (₁ ²₁₂) + (k) = (²+¹) (k + 1). Prove that r - 2 (k) + ( k + ¹ ) + ( k + ²) +…

A: We will use given formula and value of nr . nr=n!r!×n-r!

Q: For 1 ≤p ≤ ∞, find the closer of Coo in P. (Hint: It may be practical to separate into two cases, 1…

A: We have to find the closer of C00 in lp

Q: π Let (x) be the number of prime numbers in the range from 2 to x. Select the pair of inequalities…

A:

Q: 8. On which of the following intervals is (x – 1) (x + 4)(x +2) < 0? A {x|-4 < x <-2} B {xl-4,<x<–2}…

A: Given that: (x-1)2(x+4)(x+2)&lt;0

Q: 8. Find a number o such that if 0 < x − 2| < 8,then | 3x − 1| < &,where s=0.06.

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Q: Suppose xo. X1....., are distinct numbers in the interval [a,b] and f € C**[a,b). Then, for each x…

A: Suppose x0, x1, . . . xn are distinct numbers in the interval  and f∈Cn+1a, b. Then for each x in a,…

Q: (b) If z(m) denotes the factorial function of order n then show that A"z{m) = n!h" where h is…

A: b)Given,  x(n) denotes the factorial fucntion of order n.That…

Q: (a) Prove that for n ≥ 2, 3 > 2n + n². (b) For any n ≥ 1, the factorial function, denoted by n!, is…

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Q: Prove that for x > <1+ ax, if 0 < a < 1, 21+ ax if 1 < a <∞o.

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Q: Let x(k) = ku(k) h(k) = {0, ( 1, 0≤ k ≤ 4 elsewhere Let N = 11. (a) Plot x(k) for 0 ≤ k ≤ N-1. (b)…

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Q: uppose that S is the set of successful students in a classroom, and that F stands for the set of…

A: Given that expression we have to find the value of

Q: 21. Let x = R". Show that ||x||∞ ≤ ||×||2.

A: The given problem is to show the given norms identity for x belongs to Rn , ||x||infinity&lt;||x||2.

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

A:

Q: 4. Let n be a positive even integer. (a) a for all æ E {0, 1}"? Justify your answer. How many…

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Q: 12. Determine g(G) and c(G) for G = K 2k,4k for a positive integer k.

A: Graph G=Kk,2k,4k denote a tripartite Graph. A graph in which the vertices are partitioned into three…

Q: 6. Show that =+ E-(-1)"(n+ 1) E)". when Iz – 2| < 2.

A: Seriese expansion

Q: If X is uniform on [-2,2] then p(x <3) = 3/4 O O o o

A: Given  X~Uniform(-2,2) Pdf of X is  f(x)= 1/4     , -2&lt;x&lt;2       = 0         , otherwise

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

A:

Q: (7) Let S = {1- (-)" | n € N}. Determine inf S and sup S. Justify. n

A: To find the supremum and infimum of the sequence S .

Q: 6. Show that when 0 < |z – 1| < 2, Σ (z-1)(z-3) 00 --Σ n=0 = (z-1)" 2n+2 1 2(z - 1)*

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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: - Consider the open statements: P(x):x = -2 and Q(x):x² = 4 over the domain S = {-2,0,2}. State each…

A: Since you have posted a multi subparts question according to guildlines I will solve first three…

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

A: k and n are :Two non-negative integers.

Q: 3. Prove that eª >1+x for x > -1

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Q: 4. a) Suppose that for some specific problem, heuristic function h is admissible. Is h? admissible?…

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Q: Evaluate the following sums: (i) Σ(k + 1)x*, |x| < 1;

A: ∑k=0∞k+1xk,      x&lt;1 Note: There is three different question, according to the guidelines we are…

Q: Determine the global extreme values of the f(x, y): = 12x − 3y if y ≥x-4, y ≥-x-4, y ≤ 7. (Use…

A: Note: To extremize a function of two variables over a closed bounded region, we should first find…