MATH222 A1 solution

MATH222 A1 solution - MATH 222 Calculus 3 Fall 2011...

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MATH 222, Calculus 3, Fall 2011 Solutions to Assignment 1 1. For each of the following sequences ( a n ), state whether it converges or diverges. For each one that converges, ﬁnd the limit. (a) a n = cos( π 6 + 1 n 2 ) . Solution: Since the cosine is continuous, lim n →∞ a n = cos(lim n →∞ ( π 6 + 1 n 2 )) . The limit inside the bracket is π 6 + lim n →∞ 1 n 2 = π 6 . Hence a n cos( π 6 ) = 3 2 (as n → ∞ , of course). (b) a n = 1 n 2 - 2 - n 2 + 2 n . Solution: We rationalize the denominator By multiplying top and bottom by n 2 - 2 + n 2 + 2 n . We see that a n = n 2 - 2 + n 2 + 2 n - 2 - 2 n = - 1 2 "s n 2 - 2 ( n + 1) 2 + s n 2 + 2 n ( n + 1) 2 # = - 1 2 (1+1) = - 1 . (c) a n = 2 n 50 n 2 . Solution: This one diverges. It is not hard to see that for large enough n , 2 n > 50 n 3 and hence a n > n for large n . a n → ∞ . (d) a n = (ln n ) 2 n . Solution: Let

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MATH222 A1 solution - MATH 222 Calculus 3 Fall 2011...

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