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

Q: Find P (-1.5 < or equal to Z < 0.2) where Z ~ N (0,1)

A: We will use Standard Normal table to find the required probability.

Q: Let ao 1 and an 2an-1 + 1 for every n ≥ 1. Let G(r) be the generating function of the sequence ao,…

A: Introduction: The generating function is a power series representation of a recurrence relation.…

Q: 9. Establish the Bernoulli inequality: If 1 + a > for all n ≥ 1. (1+a)" 0, then 1+ na

A:

Q: 3. Assume the sequence of positive numbers (sn) increases monotonically to infinity. Prove that Sn+1…

A: Let Sn be a sequence of positive numbers sn increase monotonically to infinity. ⇒limn→∞sn=∞     ∵ sn…

Q: Solve the following recursion relation T(n) = n* T(n-1) if n>1 = 1.…

A: Explanation of the answer is as follows

Q: Show that for all x>0, e* >1+x+÷ 2.

A:

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: Show that it is not true that for every error 1/m there exists n such that |Ek=1 polt – r/(1 - r)| <…

A: We will show that for every m there does not exist any unique n that works for all r in (0,1)

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: S 6(1 – y), 0<x <y<1 (.) - {0, elseubere x, Y elsewhere

A:

Q: (1.3) Show that n - n (n – k) . k + (k + 1) k k 1 - for any two positive integers k and n with k <…

A: Let's find.

Q: Find a closed formula for S(n, n − 2), for all n ≥ 2.

A: Find a closed formula for S(n, n − 2), for all n ≥ 2.

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

A:

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

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…

A:

Q: 15. Let z = 2cos0 + i(1 – 2sin0) where -n < 0 < n Show that Iz – il = 2

A:

Q: (b) For any n ≥ 1, the factorial function, denoted by n!, is the product of all the positive…

A:

Q: Let X1 ∼ N(0.7,1.4), X2 ∼ N(−0.5,1.2), X3 ∼ N(0.9,1.7), and X4 ∼ N(−0.2,2.1), independently. 1.…

A: X1 ∼ N(0.7,1.4), X2 ∼ N(−0.5,1.2), X3 ∼ N(0.9,1.7), and X4 ∼ N(−0.2,2.1), independently.

Q: (a) Let n €N\{0}, and k e Z. Prove that k-

A: We will use very basic approach for this question in such a manner that it will enter in your head…

Q: 2. (a) Show that k!n-k-1 = for n>0 and ka nonnegative integer. 5. e-nxxk dx

A:

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: Find Pr (|w| < 0.2)  where W ~ N (-1,4)

A: From the given information, W follows normal distribution with mean µ=-1 and variance σ2=4.

Q: Prove the following inequality (1+z)" > 1+ nx + n(n 120 for any n2 1. Show that for all positive…

A:

Q: that for n > k we Sn+1 ||

A: The expression given is ,  n+1k+1=∑j=knnn-jjk The number of ways to partition n+1 objects into k+1…

Q: x[n] y[n] Σ X) ay[n – 1] D yln-11

A:

Q: Prove that 1 (c) 2F₁ (a, b; c; z)

A:

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: Prove that for n ≥ 1: 1k² = n(n + 1)(2n + 1).

A: Given that  to prove for n≥1, ∑k=1nk2=16n(n+1)(2n+1)

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: For which of the following subsets of R does a greatest lower bound in R NOT exist? I. A = {x € R :…

A:

Q: k* for some constant k". Lp.) Show that L(po)

A:

Q: II. Let U = R I= {x €R|x - 14x + 45x = 0 J= {x €R|x s6} S = {x €R| |2x| > 2}. } Find the following.…

A:

Q: 3. Prove that eª >1+x for x > -1

A:

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

A:

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: 1) Consider the conquer and divide relation f(n) = 2f (n / 4) +9 with f(1) = 3. a. Considering only…

A: Here we try to find the pattern so that we can find the general formula.

Q: 8. Let w, z EC such that |w| > |z|. Prove that (n+1) | ²7. < W for all n E N. |w| |w| - |z|

A: To prove (n+1)|zw|n&lt;|w||w|−|z| for all n∈N, it can use mathematical induction.Base Case (n =…