MidSoln - MATH 214 A1 - Midterm Monday, May 30, 2011 -...

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MATH 214 A1 - Midterm Monday, May 30, 2011 - 10:30 - 11:40 am (1) Determine whether the following sequences converge or diverge. If they converge, find the value of the limit. If they diverge, prove that they diverge. (a) a n = n 3 +ln( n ) - n 3 n +1 sin( n )+2 n +5 The sequence converges. To see this, divide both top and bottom by 2 n to get n 3 2 n + ln( n ) 2 n - n 3 n +1 2 n sin( n ) 2 n + 1 + 5 2 n analyzing each term separately using l’Hopital’s rule and applying the limit laws gives that the sequence converges to 0+0+0 0+1+0 = 0 1 = 0 (b) b n = n q ( 3 7 ) n + 1 The sequence converges. This can be proven using the sandwich theorem: 1 b n n 2 and lim n →∞ n 2 = 1 from class. (2) Find the exact value of the following series: (a) n =0 2 2 n +1 +3 n - 1 5 n - 1 The series breaks up as X n =0 10 ± 4 5 ² n + 5 3 ± 3 5 ² n = 10 1 - 4 5 + 5 3 1 - 3 5 = 325 6 (b) n =0 3 (3 n +2)(3 n +5) Using partial fractions, the series is equal to X n =0 1 3 n + 2
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MidSoln - MATH 214 A1 - Midterm Monday, May 30, 2011 -...

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