3. Choose one of the following two propositions: n (a) Prove that Vn € Z such that n ≥ 1, i *i! = 1 * 1! + 2 * 2! + ... + n * n! = (n + 1)! − 1. i=0 (b) Prove that Vn EZ, n³ +n is even.

Q: Prove that Eo 2i = 2n+1 – 1 - i3D0

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Q: Let S={1/n-1/m: m,n ∈ Z+} What are infS and supS? prove your answers. (you can use the Archimedean…

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Q: 2n+2 52n+1 3n+2, 22n+1 is divisible by 19 co dady M N [ day f04+ 204d (21)

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Q: п 1 п V n 21 j=1 j(j+1) (п+1)

A: From the given information. The objective is to prove the following. ∑j=1n1jj+1=nn+1

Q: If (ar + br) = 2n %3! n+1 and k-1 8. E(ak - br) = n+2 - , then ar %3D k-1 k-1

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Q: Prove or disprove that for all n > 1 п i2' < 2"+1n. i=1

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Q: (-1)* x* k! k! k=0 k=0 2k а) 2 (k!)* k=0 b) 1 (-1)*. (2k)! k 2k k=0 d) 0 e) None of these.

A: Expansion of Exponential function

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: Prove that (n – 1) +n° + (n + 1)° is divisible by 9 for n E {1,2,3, . }

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Q: Prove that for every non-negative interger n: 2n+1 + (-1)" Σ 3· 2n i=0

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Q: Let X1, X2,... be independent. In which of the following situations does (X) satisfy the Lindeberg…

A: The Lindeberg central limit theorem, a variation of the central limit theorem that applies to a…

Q: Let A = { n - m : nEN, m EN, m > n}. (a) Prove that inf A = 0. (b) Prove that sup A = 1.

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Q: (11) 11. Prove by induction that 1 1 1+ + + 22 3² + - 2.

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Q: Let Y= E an x" ) compute y'aned y"and write out the first four terms of each sexies, as well as the…

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Q: verify that det(A B) = det(B A) and determine whether the equality det(A + B) = det(A) + det(B)…

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Q: 1. Let A=(a)xn be symmetric and positive definite. Show that a) a>0, i=1,2,...,n. |a|< (aa)², i,…

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Q: 28. Prove by induction that i(i – 1) (n+1)n(n– 1) i=1 holds Vn E N,n 2 1.

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Q: Prove that (2) (6) (10) (14) ...(4n-2) = (2n)! n!

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Q: | Vyn+z V = h ( 1 + 1/ ▼ + 5₂ V ² +.......³/₁ n 12 Prove that Vyn+1

A: Please find step by step solution below:

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: -1] and h[n]

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Q: {in(n² +1) Im(x +4)} 19. А. В. 1 С. 1/2 D. 2 E. Divergent

A: we have to identify the given sequence is convergent or divergent

Q: Show that os fj ex* +en+ 22n+1 + (2n+1)n! 25 e*(22n+3) dx – |2 + ... 5(2!) (n+1)!

A: The given integral is ∫02ex2dx. Using Taylor series, the exponential function can be written as:…

Q: (17) Define S(n) as follows: n S(n) : i(i!) = (n+ 1)! – 1 i=1 Prove Vn e N1, S(n).

A: Consider the given expression. Sn=∑i=1nii!=n+1!-1 Now, use the mathematical indication to prove the…

Q: Use Mathematical Induction to show that, for all n € Z+, 6 (10" + 2). Suppose z = (Inn-1...1x0)10…

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Q: Let A = {: : neN, mN, m >r m (a) Prove that inf A 0. (b) Prove that sup A = 1.

A: Given that a set  A=nm: n,m ∈ℕ, m&gt;n (a) Prove that infA=0. (b) Prove that supA=1.

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: is the condition such that the straight lines joining the origin to the other two points of…

A: Given: Curves equation   ax2+2hxy+by2+2gx=0a'x2+2h'xy+b'y2+2g'x=0

Q: S = {x2 − y2 ∈ N : x, y ∈ Z, x2 − y2 > 127}. Prove that S has a smallest element.

A: Given S={x2-y2∈N: x,y∈Z, x2-y2&gt;127} Theorem (Well ordering Principle) Every Nonempty subset of…

Q: 4. Show that P(n,k)= P(n – 1, k)+ k · P(n – 1,k – 1). Using Combinatorial Proof.

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Q: Is the stries cm vergent or divergent? (Ghow how you kenowi) 2 3n n'+ In + 13

A: a The given series is ∑n=1∞23n.We know that the series ∑n=1∞1n is divergent.And…

Q: 5. Use the binomial theorem to prove that C(n, 0) C(n, 1) + C(n, 2) + (-1)"C(n,n) = 0. —

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Q: Let A = { n - m : neN, me N, m > n}. (a) Prove that inf A = 0. (b) Prove that sup A = 1.

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Q: (5a) Let A = {3, 4} be a subset of S = {1,2,... ,6} and let n E S. || n2(n+1)² Prove that if is…

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Q: Prove the following using Ito's lemma, n f 0 B2n = 2n B2n-¹dB, + n(2n-1) •t 0 B²(n-1) ds.

A: We need to prove that: Bt2n=2n∫0tBs2n-1dBs+n2n-1∫0tBs2n-1ds. We will be denoting Bs as standard…

Q: 4. Determine whether or not the following is true: n+1 (a) i2i = n. 2n+² + 2, Vn ∈ N, n>1. (b) i=1…

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Q: Consider these four statements: 1 A: 2 + 23+... +...+ 2.3 for every integer n > 1. n(n+1) n+1' B:…

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