11) Assume that for an infinite family of eens we have that An+1 C An for...
11) Assume that for an infinite family of events (Ann1 we have that AntI C An for any neN and that Show that 1 of 1 you don't have to give a rigorous topological proof. P(A)lm (A) -+00 r prokm
An important fact we have proved is that the family (enr)nez is orthonormal in L (T,C) and complete, in the sense that the Fourier series of f converges to f in the L2-norm. In this exercise, we consider another family possessing these same properties. On [-1, 1], define dn Ln)-1) 0, 1,2, Then Lv is a polynomial of degree n which is called the n-th Legendre polynomial. (a) Show that if f is indefinitely differentiable on [-1,1], thern In particular,...
(Limit of functions) Let f : 2-» C be a function, and assume that D(a, r) C Q. We say that lim f(z) L Ď(a, 6) we have |f(z) Ll < e. if for any e > 0 there exists 6 > 0, such that for any z e (a) State the negation of the assertion "lim^-,a f(z) = L". (b) Show that lim- f(z) L if and only if for any sequence zn -» a, with zn a for...
Answer True or False and explain
1 The infinite family {Pn(x)}^=o of Legendre polynomials Pn(x) forms a complete orthogonal family on the interval [-1, 1]. If we delete the first element Po(x) = 1 from the set, the remaining family {Pn(x)}=1 also forms a complete orthogonal set. 2 Let {Xn}n=1 be a complete orthogonal family of functions for the vector space L[0, 1]. Then enlarging the set by adding to this set the vector 2X5 + 3X18, we end up...
1. Verify that the following linear system does not have an infinite number of solutions for all constants b. 1 +39 - 13 = 1 2x + 2x2 = b 1 + bxg+bary = 1 2. Consider the matrices -=(: -1, -13). C-69--1--| 2 -1 0] 3 and F-10 1 1 [2 03 (a) Show that A, B, C, D and F are invertible matrices. (b) Solve the following equations for the unknown matrix X. (i) AXT = BC (ii)...
11. As shown in the below figure, we connect an infinite number of capacitors with the value C =1 Farad. What is the equivalent capacitor of the system between points K and L?
2. (a) For 0<x+1, we have (x - 1)/x <Ins<:- 1. (b) For je N. > 1, we have in(+1)<1/; <In(;). (c) For n,ke N, n > 1, we have In(k+)< } <In(a + and lim . = Ink (d) Use (c) to prove that ) - 1) * = In 2 (Hint: Show that
The exponential function V-e increases on the interval The logarithmic function y = ln ( x) increases on the interval By definition, In(e) Hence, for all x >0 it follows that Ine-1)< In(e-1 and we immediately have thatx201x0 for all x>0 2.01 Since is a p-series with p- /n In (e"-1 by direct comparison, we conclude that 2.01 3b: Complete the outline to verify the convergence or divergence of the infinite series using limit comparison. In(e-1 and b" and then...
Claim: {(-1)"} does not converge to any real number a. Proof: Assume that the sequence converges; that is, assume that there is an a E R such that lim,--.(-1)" = a. Then, using E = 1, from the definition of convergence, we know that there exists an no such that |(-1)" - al < 1 for all n > no. Thus, for any odd integer nno, we have |(-1)" - al = 1-1-a[< 1, and for any even integer n>...
3. Assume we have Simpson's Rule: to = a, 13 = , h = (b-a)/2 = a +h. (20) + 47(01) + f(x)]- ()where do < < Let fe .b), be even, h= (b-a)/n, and = a + jh, for each j = 0,1...... Show that there exists a l E (a,b) for which the Composite Simpson's rule for n subintervals can be written with its crror term as n/2 bar (n/2) - 1 f(a) +2 =1 (12) + 4...