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Solutions1

# Solutions1 - MATH 214 A1 Assignment 1 Solutions Due Friday...

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MATH 214 A1 - Assignment 1 Solutions Due Friday, May 13, 3:00pm (1) Do the following sequences converge? If so, find the limit, if not, prove that they do not: (5 marks each) (a) a n = 3 n 4 +2 n 2 +5 n - n n 4 +3 n - 2 Converges: Use either l’Hopital’s rule or divide both top and bottom by n 4 and use limit rules. (b) b n = n cos ( 1 n ) Diverges: If b n converges, then b n cos ( 1 n ) = n converges, which is clearly false. (2) Suppose that f ( x ) is differentiable for all x [0 , 1] and that f (0) = 0. If a n = nf ( 1 n ) : (a) Show that lim n →∞ a n = f 0 (0)(5 marks) By definition: f 0 (0) = lim h 0 f (0+ h ) - f (0) h = lim h 0 1 h f ( h ). Using that lim h 0 g ( h ) = lim n →∞ g ( 1 n ) and the ”Connect the dots” the- orem from class, lim h 0 1 h f ( h ) = lim n →∞ nf ( 1 n ) (b) Use part a to calculate the limit of b n = n sin ( 1 n ) (2 marks) Using f ( x ) = sin( x ), we see that lim h 0 b n = f 0 (0) = cos(0) = 1. (3) Does n =1 n sin ( 1 n ) converge? If so what is the value? If not, why not? (3 marks, Hint: Look at b n in question 2) No. lim n →∞ b n = 1 6 = 0 (4) Determine whether the following series converge or diverge. If

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Solutions1 - MATH 214 A1 Assignment 1 Solutions Due Friday...

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