(a) Let ai=2i . Find ∑20i=1 ai+1-ai . (b) Let bi=i2. Find ∑20i=1 bi+1-bi .
Problem 3. Fix any p > 1. Let ai, ..., An and bi, ..., bn be real num- bers. Prove that 1/ /n 1/P > ja, + 6P ( * wr)" (£wr)" (Mr) P + 1 (Hint: Minkowski inequality for a proper probability space.)
2. Let 6 marks (a) Find f(x),f"(x), and f"(x). (b) Find the second order Taylor expansion of f at 1, namely f(r) = ao + ala-1 ) + a2(z-1)2 + R2(x), where Ra is the remainder. You should find ao, a, a2, and R(p). 8 marks that the error in this estimation (i.e., R2(0.9)1) is at most 10-3. 6 marks (c) Use the Taylor expansion found above to estimate the value of f(0.9). Show Find f(x), f"(), and f" (b)...
Let f (x) be a monic polynomial of degree n with distinct zeros ai,..., an. Prove -1 Let f (x) be a monic polynomial of degree n with distinct zeros ai,..., an. Prove -1
4. Let A be a square matrix such that AAT-1. Show that AI = ±1. 4. Let A be a square matrix such that AAT-1. Show that AI = ±1.
1: We define the Vandermonde Determinant, denoted V(ai,a2,... ,an), as ai a...a-1 1 2 i a2 az...a-1 al,a2 ,an ) 2. 1 an a an ...an-1 We will guide you through a proof by Mathematical Induction to show that V(a,aan) aj -ai f: Show that if we perform k Type 3 ccolumn operations by adding a multiple B, of col- umn i, where1,2,. ,k, to the last column, then the Vandermonde determinant of size (k 1) x (k 1) can...
Let Ai,i= 1,2,· · ·, are events such thatP(Ai) = 1 for all i. Prove thatP(⋂∞i=1Ai) =1.
1) Consider the switching networks shown in Fig. 1. Let Ai, A2, and As denote the events that the switches s1, s2, and s3 are closed, respectively. Let Aab denote the event that there is a closed path between terminals a and b. Express Aab in terms of Ai, A2, and A3 for each of the networks shown. 2 Figure 1
5*. Consider all sequences (ai,. .., an) such that a, are nonnegative integers and a ai+ 2. Let P, n and Rn be the number of such sequences which start from 0, 1 and 2 respectively. (a) Compute P, Qn, Rn by writing down all such sequences for n 1,2,3. (b) Prove that P, Qn Rn satisfy the recurrence relations: (c) Translate the above equations into linear equations for the generating functions for P, Qn, Rn (d) Solve these equations...
Question 6: Let n 2 2 be an integer and let ai,a2,...,an be a permutation of the set (1, 2, . . . ,n). Define ao = 0 and an+1 = 0, and consider the sequence do, 1, d2, l3, . . . , Un, Un+1 A position i with 1 i n is called auesome, if ai > ai-1 and ai > ai+1. In words, i is awesome if the value at position i is larger than both its...
pigeonhole 1. Show that in every sequence (ai,a2, a100) of the letters A,B,C,D, there are two indices 1i< j < 98 such that (ai, ai+1,aj+2) = (aj, aj+1, aj+2).