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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...
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 comple...
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)...
(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...
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...
If we have the assumption that the residues at individual positions in a family of DNA...
If we have the assumption that the residues at individual positions in a family of DNA sequences are independent and have the identical probabilities, what would be the maximum likelihood estimates (MLE) of the probabilities of having different bases at each residue in this family: P(A), P(G), P(C), and P(T) if we have already known that the following five sequences belong to this family? P(A) = 0.03, P(G)=0.6, P(C) = 0.3, P(T)=0.07 CGGTGCG; ACGGAGCG; GCGGTGTTGG; AACCGGG; ATCGGTGG
1. Verify that the following linear system does not have an infinite number of solutions for...
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...
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. >...
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) incre...
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;...
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>...
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>...
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