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A08 - terms prove that s-s 6 ≤ 1 729 4 Determine whether...

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Math 138 – Fall 2011 Assignment 8 Due Monday Nov 14 (in the drop box by 12 noon) Hand in the following: 1. Determine whether or not the following series converge or diverge by making a suitable comparison. (a) X n =1 n n 2 + 2 n + 2 (b) X n =1 sin 1 n (c) X n =1 3 1 /n n 3 (d) X n =1 arctan n n (e) X n =1 1 + 1 n e n (f) X n =1 1 n 1+ 1 n 2. (a) Show that ln x x 1 / 3 0 as x → ∞ . ( this is not a ‘prove using the definition’ question...just a simple calculate the limit. You may want to review L’Hopital’s rule in Section 4.4 of the text.) (b) From part (a) we can thus say that the sequence ln n n 1 / 3 1 con- verges to 0. Use the definition of convergence with = 1 , to deduce that for sufficiently large n , (ln n ) 3 n .
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(c) Show that the series X n =2 1 (ln n ) 3 diverges. 3. (a) By making a comparison to a certain geometric series, show that X n =1 1 + sin n 1 + 3 n converges. (b) If s is the sum of the above series and s 6 is the sum of the first 6
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Unformatted text preview: terms, prove that s-s 6 ≤ 1 729 . 4. Determine whether or not the following series are convergent. (a) ∞ X n =2 (-1) n ln n (b) ∞ X n =1 (-1) n ( √ n 2 + 2 n-n ) (c) ∞ X n =1 (-1) n +1 e 1 /n n 5. Explain why each series below converges. Then find the smallest integer n such that the n th partial sum s n will approximate the actual sum of the series s with error less than the given bound. (a) ∞ X n =1 n (-1) n-1 n 2 + 1 , with error less than 10-3 (b) ∞ X n =1 (-1) n +1 n 2 2 n , with error less than 10-3 . (c) ∞ X n =1 (-1) n ( n !) 2 , with error less than 10-6 . 6. Does the series ∞ X n =0 = √ 2 sin ± (2 n + 1) π 4 ² cos ³ nπ 2 ´ (-1) n ( n + 1)-1 converge or diverge. Justify your answer. 2...
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