211-2007&2008-2-M10-April2008

211-2007&2008-2-M10-April2008 - K u w a i t U n i v...

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Unformatted text preview: 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] 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 =    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 ∑ ∞...
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This note was uploaded on 02/23/2010 for the course CAL C 0410211 taught by Professor Deparpment during the Spring '10 term at Kuwait University.

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211-2007&amp;2008-2-M10-April2008 - K u w a i t U n i v...

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