cheatsheet1B - 2 , (0, 1) 3, 1 2, 4, 2 2 2, 2 , 23 , 1 6 2...

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π 2 , (0 , 1) 0 , (1 , 0) π 6 , 3 2 , 1 2 ± π 4 , 2 2 , 2 2 ± π 3 , 1 2 , 3 2 ±
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Squeeze theorem: if a n b n c n for n N and a n L and c n L then b n L . If lim n →∞ | a n | = 0 then lim n →∞ a n = 0. If n =1 a n is convergent, then lim n →∞ a n = 0. Integral test: if a n = f ( n ) for n N and f ( x ) is continuous, positive and decreasing for x N , then ( a n is convergent) (there is an x 0 such that x 0 f ( x ) dx is convergent) Comparison test: Suppose 0 a n b n for n N . Then (1) b n convergent a n convergent. (2) a n divergent b n divergent. Limit comparison test: Consider the series a n and b n . Suppose b n > 0 for n N . If lim n →∞ a n b n = c , where 0 <c< , then either both series converge or both diverge. Alt. series: if 0 b n +1 b n for n 1 and b n 0 as n →∞ , then ( 1) n b n converges. estimation: | s s n | <b n +1 , where s = n =1 ( 1) n b n and s n = n i =1 ( 1) i b i .
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This note was uploaded on 04/04/2010 for the course MATH MATH 16B taught by Professor Unknown during the Spring '10 term at UC Davis.

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cheatsheet1B - 2 , (0, 1) 3, 1 2, 4, 2 2 2, 2 , 23 , 1 6 2...

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