4. [10 points) Prove the following 0, m n .cos mar cos nr olz =İl, mı...
(-1)-1 n2 is absolutely convergent. 1. (2 points) Prove that cos n is convergent or divergent. 2. (2 points) Determine whether the series - (Use cos n<1 for all n) 3. (3 points) Test the series -1) 3 for absolute convergence. (Use the Ratio Test) 2n +3) 4. (3 points) Determine whether the series converges or diverges. 3n +2 n-1 (Use the Root Test) 5. (3 points) Find R and I of the series (z-3) 1 Find a power series...
2. For n . define functions T inductivelv such that 0, 1, 2, . . . (cosx) = cos(nx), with Folz) 1. (a) Prove that Tn is a polynomial for every n and compute its degree. b) Prove the recursion formula (c) Compute the integral dr 山 for every n, m E N
2. For n . define functions T inductivelv such that 0, 1, 2, . . . (cosx) = cos(nx), with Folz) 1. (a) Prove that Tn is...
VER, DER, 4) Prove that the rotation matrices [cos – sin 07 1(0) 4 sinŲ cos x 0, 0 0 1 cose 0 sin 0] O(0) 4 0 1 0 , 1-sin 0 cos e ſi 0 0 1 0(0) 4 0 cos – sin 0, 0 sinº cos 0 ] are rotation matrices, that is, V-7(4) = \T(4), 6-7(0) = OT(0), $ER, 6-7(0) = $1(0), and det(\())) = 1, det(O(0)) = 1, det($(0)) = 1. Prove also that R321(4,0,0)...
4. (15 points) Prove or disprove each one of the following statements: ·0(f(n) + g(n)) = f(n) + 0(g(n))I f(n) and g(n) are strictly positive for all n ·0(f(n) × g(n)) positive for all n f(n) 0(g(n))I f(n) and g(n) are strictly
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2. (10 points) Let R be an integral domain and M a free R-module. Prove that if rm 0 or m 0 where r E R and m E M, then either r 0.
2. (10 points) Let R be an integral domain and M a free R-module. Prove that if rm 0 or m 0 where r E R and m E M, then either r 0.
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...
n=0 4. Using the power series cos(x) = { (-1)",2 (-0<x<0), to find a power (2n)! series for the function f(x) = sin(x) sin(3x) and its interval of convergence. 23 Find the power series representation for the function f(2) and its interval (3x - 2) of convergence. 5. +
Q2 (10 points) Vn2 + 4 – n, n E N. 2. Let (an) neN be the sequence with a, (a) Prove that lim,→0 an 0. lim,-00 bn, and prove the limit exists, by using the definition. (b) Let bn = n an . Find L =
NR HC Ki AgNO; Na2CO3 Naci Na2SO4 H2SO4 BaCl2 NR NR PPT PPT N R PPT PPT HC NR NR NR PPT NR | NR NR INR | KI - NR NR PPT NR NR NR NR AgNO3 PPT NR PPT PPT PPT PPT Na2CO3 NR PPT NR NR NR NR NaCl NR | NR | PPT NR NR NR NR Na2SO4 NR NR PPT NR NR | NR NR H2SO4 NR NR PPT NR NR NR NR 1. Write...