Solutions3 - MATH 214 A1 - Midterm Solutions (1) Find the...

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MATH 214 A1 - Midterm Solutions (1) Find the limits of the following sequences or show they diverge: (a) { a n } n =1 = { ( 1 + 1 n ) 1 n } n =1 (b) { b n } n =1 = { n n n 2 n } n =1 Solution. (a) We have that 1 ± 1 + 1 n ² 1 n 2 1 n for all n . We also know that lim n →∞ x 1 n = 1 for all x > 0 from class, so by the sandwich theorem, lim n →∞ a n = 1 (b) For every n , 1 n 1 n - 1 n 2 n 1 n so lim n →∞ b n = 1 by the sandwich theorem. ± (2) For each of the following series, identify whether they converge absolutely, converge conditionally or diverge and prove your answer: (a) n =1 ( 1 + 1 n ) 1 n (b) n =2 ( - 1) n n 2 - 1 Solution. (a) This series cannot converge because the terms of the sequence do not converge to 0 (as proven in 1 a ). (b) To determine absolute convergence, notice that 1 n 2 - 1 1 n = n n 2 - 1 1 as n → ∞ so the sequence does not converge absolutely by the limit comparison test applied to the harmonic series. Let
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Solutions3 - MATH 214 A1 - Midterm Solutions (1) Find the...

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