exam1_sl - M408D Midterm Exam 1 57075/80/85 September 16 1....

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57075/80/85 September 16 1. True or False (4 points for each): (1) If a n 0 , a n +1 a n for all n N and lim n →∞ a n = 0 , then X n =1 a n is convergent. (2) If lim n →∞ ( a n +1 - a n ) = 0 , then the sequence a n is convergent. (3) Suppose the series X n =0 a n is convergent and the series X n =0 b n is divergent. Then the series X n =0 ( a n + b n ) is also divergent. (4) By L'Hospital's rule, we have lim x 0 sec x 1 - cos x = lim x 0 sec x tan x sin x = lim x 0 1 cos 2 x = 1 . (5) Although two series X n =0 a n and X n =0 b n are both divergent, the series X n =0 ( a n + b n ) could be convergent. Answer (Mark T for True and F for False): (1) F. For example, a n = 1 /n . (2) F. For example, a n = n X k =1 1 k . Then a n +1 - a n = 1 / ( n + 1) . (3) T. Proof: By contradiction. Let c n = a n + b n and suppose X n =1 c n is convergent. Then by the Theorem 8(on page 729 in the textbook), X n =1 c n - a n is also convergent. Since c n - a n = b n , it implies X n =1 b n is convergent. This contradicts to X n =1 b n is divergent. (4) F.
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exam1_sl - M408D Midterm Exam 1 57075/80/85 September 16 1....

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