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review2 - MAC 3473 FALL 2007 REVIEW PROBLEMS FOR TEST 2 1...

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MAC 3473 - FALL 2007 REVIEW PROBLEMS FOR TEST 2 1 . Determine whether each of the following integrals is convergent or divergent. Be sure to show your work and explain your reasoning. a) Z 5 dx x (ln x ) b) Z 2 x 14 + 1 x 15 - x dx c) Z 4 1 dx 4 - x d) Z 1 0 ln x x dx 2 . Determine whether the following improper integrals convergent or divergent. Be sure to show your work and explain your reasoning. a) Z 0 dx ( x + 1)( x + 3) b) Z 0 dx (4 + x 2 ) 3 / 2 c) Z 1 0 dx (1 - x 4 ) 1 / 12 3 . (i) Define what it means for the number L to be a limit of the sequence { a n } n =1 . (ii) Prove the following limits using the formal definition. a) lim n →∞ 1 3 n + 1 = 0 , b) lim n →∞ 3 n - 1 2 n + 1 = 3 2 . (iii) Determine whether the following sequences converge, giving reasons. Find limit if it exists. a) a n = ( n + 1)( n + 2) ( n + 3)( n + 4) b) b n = ( - 1) n cos(1 /n ) c) c n = n + 2 - n - 1 d) d n = n e n e) e n = (3 n ) 1 /n (iv) State the Squeeze Theorem. (v) Use the Squeeze Theorem to prove that lim n →∞ e cos n n = 0 .
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