sample_midtermI_21c2

# sample_midtermI_21c2 - (a 1-1 2 2 1 3 2-1 4 2 1 5 2-1 6 2 1...

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Calculus Math 21C, Fall 2010 Sample Questions: Midterm I 1. Do the following sequences { a n } converge or diverge as n → ∞ ? Give reasons for your answer. If a sequence converges, ﬁnd its limit. (a) a n = cos n n ; (b) a n = n ln n ; (c) a n = n 2 + 1 - n. 2. Do the following series converge absolutely, converge conditionally, or diverge? Give reasons for your answer. (a) n =1 ( - 1) n +1 n ; (b) n =1 sin n n 2 ; (c) n =1 ( - 1) n sin n 3. Determine whether each of the following series converges or diverges and explain your answer: (a) n =1 n + 4 6 n - 17 ; (b) n =1 ( - 4 5 ) n ; (c) n =2 n n 4 + 7 ; (d) n =1 5 n +1 (2 n )! ; (e) n =3 1 n ln 2 n ; (f) n =3 ( - 1) n +1 n + 2 n ; (g) 1 1 4 + 1 2 4 - 1 3 4 + 1 4 4 + 1 5 4 - 1 6 4 + 1 7 4 - 1 9 4 + ··· ; (h) n =1 ( n !) 2 (2 n )! ; (i) n =1 [tan( n ) - tan( n + 1)] . 1

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4. Are the following equalities true or false? Justify your answer.
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Unformatted text preview: (a) 1-1 2 2 + 1 3 2-1 4 2 + 1 5 2-1 6 2 + 1 7 2 + . . . = 1 + 1 3 2-1 2 2 + 1 5 2 + 1 7 2-1 4 2 + 1 9 2 + 1 11 2-1 6 2 + . . . ; (b) 1-1 2 + 1 3-1 4 + 1 5-1 6 + 1 7 + . . . = 1 + 1 3-1 2 + 1 5 + 1 7-1 4 + 1 9 + 1 11-1 6 + . . . ; 5. State the deﬁnition for a sequence { a n } to converge to a limit L . If a n = n 2 + 1 n 2 for n = 1 , 2 , 3 , . . . prove from the deﬁnition that lim n →∞ a n = 1 . Additional question. Does the series ∞ ∑ n =2 1 (ln n ) ln n converge or diverge? Justify your answer. 2...
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sample_midtermI_21c2 - (a 1-1 2 2 1 3 2-1 4 2 1 5 2-1 6 2 1...

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