solution :
7)By using Stirling's formula
8)By induction rule :
7. The last exercise in the handout entitled Some Common Functions. Use Stirling's formula to prove...
1. Prove the following for the normalizing factor of the t-density - Hint: Use Stirling's formula as 0o.
6) This question is part of Exercise 3-4 on page 62 of CLRS, but the letters aren't all the same. Answer TRUE or FALSE for each of the following statements. TRUE means that the statement is TRUE for any asymptotically positive functions f(n) and g(n). Otherwise, answer FALSE. You don't have to Prove or Disprove these statements... but you should learn how to do that. f(n)- O( g(n)) implies g(n) O( f(n)) b) This is part a in Exercise 3-4....
4. Stirling's Formula is the claim that n! n-o0 >1. V2nn(n/e)" In this exercise, we will show how this can be obtained from the Central Limit Theorem Recall that Exponential () if fx(x)= Ae ^x, x > 0; the corresponding mgf is Mx (t) ,t<^ X = and GA)".,1 ва xa-le-px, x> 0; the corresponding mgf is Mx(t) = X~T(a,B) if fx(x)= T , t <B (a) Argue that, if Xi ~Exponential(1), i = 1,2,..., all independent, then for every...
5. Use mathematical induction to prove that for n 2 1, 1.1! +2.2!+3.3++ n n! (n +1)!-1 7. Prove: If alb and al(b +c) then alc. Prove that for all sets A and B, P(An 6. 8. (a) Find the Boolean expression that corresponds to the circuit 5. Use mathematical induction to prove that for n 2 1, 1.1! +2.2!+3.3++ n n! (n +1)!-1 7. Prove: If alb and al(b +c) then alc. Prove that for all sets A and...
Q (8 points) Use mathematical induction to prove the formula 1 X – 1 1 X x(x – 1) 22 2n for all n = 1, 2, 3, ..., and x + 0,1.
1. (Exercise 4.10, modified) Given a series Σ 1 ak with ak 0 for all k and lim Qk+1 k0oak we will prove that the series converges absolutely. (This is part of the ratio test sce the handout.) (a) Fix a valuc q with r <<1. Use the definition of r to prove that there exists a valuc N such that for any k 2 N. (b) Prove that Σο, laNIqk-1 converges, where N is the value from part (a)....
how do I prove this by assuming true for K and then proving for k+1 Use mathematical induction to prove that 2"-1< n! for all natural numbers n. Use mathematical induction to prove that 2"-1
Questions 3, 5, 7 - Mathematical Structures | 1ỏ +2° +33 ...3 - Rº(n1) for all integers n > 1. 2. Use induction to prove that the following identity holds for all integers n > 1: 1+3+5+...+(2n - 1) =n. 3. Use induction to show that for all positive integers n. 4. Use induction to establish the following identity for any integer n 1: 1-3+9 -...+(-3) - 1- (-3)"+1 5. Use induction to show that, for any integer n >...
Induction proofs. a. Prove by induction: n sum i^3 = [n^2][(n+1)^2]/4 i=1 Note: sum is intended to be the summation symbol, and ^ means what follows is an exponent b. Prove by induction: n^2 - n is even for any n >= 1 10 points 6) Given: T(1) = 2 T(N) = T(N-1) + 3, N>1 What would the value of T(10) be? 7) For the problem above, is there a formula I could use that could directly calculate T(N)?...
Exercise 2.4. Prove the two statements below:Use nd ueTion 1. For every integer n 2 3, the inequality n2 2n +1 holds. Hint: You can prove this by induction if you wish, but alternatively, you can prove directly, without induction.) 2. For every integer n 2 5, the inequality 2" n holds. (Hint: Use induction and the inequality in the previous part of the exercise.)