FinalMath2JSummer11_solutions

# FinalMath2JSummer11_solutions - Math 2J Summer 11...

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Unformatted text preview: Math 2J, Summer 11 Instructor : Yunho Kim The Final Exam Date : Sep. 7th UID Name Class Lec. B, 9:00am - 10:50am Number Score 1. a) b) 2. a) b) 3. 4. 5. 6. 7. Total Score / 100 1 Problem 1. Determine if the following series are convergent. Justify your answers . And if it is convergent, then find the value of the series. a. (10pts) ∞ summationdisplay n =1 a n , where a n = 1 n 2 + 5 n + 6 Solution. By the partial fractions method, we know that ∞ summationdisplay n =1 1 n 2 + 5 n + 6 = ∞ summationdisplay n =1 parenleftBig 1 n + 2- 1 n + 3 parenrightBig = lim k →∞ k summationdisplay n =1 parenleftBig 1 n + 2- 1 n + 3 parenrightBig Notice that the k th partial sum is k summationdisplay n =1 parenleftBig 1 n + 2- 1 n + 3 parenrightBig = parenleftBig 1 3- a3 a3 a3 1 4 parenrightBig + parenleftBig a3 a3 a3 1 4- a3 a3 a3 1 5 parenrightBig + ··· + parenleftBig a0 a0 a0 1 k + 2- 1 k + 3 parenrightBig = 1 3- 1 k + 3 . Therefore, ∞ summationdisplay n =1 1 n 2 + 5 n + 6 = lim k →∞ parenleftBig 1 3- 1 k + 3 parenrightBig = 1 3 . b. (10pts) ∞ summationdisplay n =1 a n , where a n = 2 n + (- 3) n 4 n Solution. Notice that ∞ summationdisplay n =1 2 n + (- 3) n 4 n = ∞ summationdisplay n =1 2 n 4 n + ∞ summationdisplay n =1 (- 3) n 4 n = ∞ summationdisplay n =1 parenleftBig 1 2 parenrightBig n + ∞ summationdisplay n =1 parenleftBig- 3 4 parenrightBig n and both infinite series are geometric series. Therefore, ∞ summationdisplay n =1 2 n + (- 3) n 4 n = ∞ summationdisplay n =1 parenleftBig 1 2 parenrightBig n + ∞ summationdisplay n =1 parenleftBig...
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FinalMath2JSummer11_solutions - Math 2J Summer 11...

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