For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
Prove the trigonometric identity: sin(x + y) sin(x - y) = sin’x – sin? y. Which identity is used to prove it true? sin(x + y) = sin x cos y - cos x sin y All of these. tan o sin e cos e cos? 0 = 1 - sin? 0
(2) Using the identity: n! k!(n - k)! for n > 2, prove that the following identity is even: 1 n
8. Suppose b 2 a z 0 for all k E N. a. lf 2. bconverges, prove that £f.i ai converges b. If Ea diverges, prove that £** b, diverges.
(3) Using the identity: (*) – 16–191 n! k!(n-k! k for n > 2, prove the following identity: (n-2 + (5+1) 1
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
(2) Prove that if j-0 i-0 with k, 1 e N u {0), and bo, . . . , be , do, . . . , dl e { 0, . . . , 9), such that be, de # 0, then k = 1 and bi- di fori 0,.. , k. (I recommend using strong induction and uniqueness of the expression n=10 . a + r with a e Z and re(0, 1, ,9).) (3) Prove that for all...
(Abstract Algebra) Please answer a-d clearly. Show your work and explain your answer. (a) Let G be a group of order 4 with identity e. Show that G is either cyclic or a2-e for all (b) Does the result of part (a) generalize to groups of order p2 for any positive integer p? In other words, is it the case that if G is a group of order p2 with identity e, then is either cyclic or a- e for...
2. Let K F be fields and a, β E F. Define K(a, β) to be the intersection of all subfields of F containing K, a B. Define K(a β) to be the set of all elements in F of the form pa, β)-g(a, β) 1, where P(z,y), g(z, y) are polynomials in Kr, y] and g(a,B)0 (i) Prove that K(a, B) are K(a, B) are subfields of F and that they are, in fact, the same subfield. (ii) Find...
Prove the identity, nx p nxn