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211-2007&amp;2008-2-M10-April2008

# 211-2007&amp;2008-2-M10-April2008 - Kuwait University...

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K u w a i t U n i v e r s i t y Department of Mathematics and Computer Science Math 211 April 9, 2008 Calculus (3) First Midterm Time: 90 minutes 1. Determine whether the sequence { a n } converges or diverges, where [2 pts] a n = 2 n + 3 3 n - 2 if n is odd 3 n + 2 2 n - 3 if n is even 2. For a fixed real number c , define the sequence { a n } by a n = 1 n c sin π n . For what values of c does this sequence converge? [3 pts] 3. Find the sum of each of the following convergent series: [5 pts] (a) X n =3 2 n 2 - 2 n (b) X n =1 3 n - 2 n ( - 4) n 4. Test the following series for absolute convergence (AC), conditional convergence (CC), or divergence (D): [9 pts] (a) X n =1 ( - 1) n sin 1 n (b) X n =1 ( - 1) n cos 1 n (c) X n =0 ( - 1) n n 2 e n 5. Find the radius and interval of convergence of the series X n =0 n + 1 2 n + 1 ( x - 1) n 2 n . [3 pts] 6. Find the Taylor series for f ( x ) = ln(1 + x ) about the point a = 1. [3 pts]

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A n s w e r s 1. Divergent since the odd terms 2 / 3, while the even terms 3 / 2. 2. Express the terms as a n = 1 n c sin π n = π n c +1 sin( π/n ) ( π/n ) and take the limit as n tends to infinity: lim n →∞ a n = lim n →∞ π n c +1 sin( π/n ) ( π/n ) = lim n →∞ π n c +1 · 1 = lim n →∞ π n c +1 = 0 if c > - 1 π if c = - 1 if c < - 1 Hence, the sequence converges for all values of c ≥ - 1. 3. (a) this is a telescoping series X n =3 2 n 2 - 2 n = X n =3 1 n - 2 - 1 n = 1 - 1 3 + 1 2 - 1 4 + 1 3 - 1 5 + 1 4 - . . . = 3 2 (b) this is a difference of two convergent geometric series n =1 3 n - 2 n ( - 4) n = n =0 3 n - 2 n ( - 4) n
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211-2007&amp;2008-2-M10-April2008 - Kuwait University...

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