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MATH 1005 (21)

# math 2004 T4Solutions.pdf

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School
Carleton University
Department
Mathematics
Course
MATH 1005
Professor
Ida Toivonen
Semester
Fall

Description
MATH 2007A Test 4 Solutions November 22, 2013 [Marks] [5] 1. Determine whether the sequence converges or diverges. If it converges, ﬁnd the limit. Justify your answers. n n 3 2 (a) 2 − 1 (b) (−2) (c) (−4) (d) n +2 n (e) n . 3n 3n 3n 3n2 en Solutio: n n 1 1 1 1 (a) lim 2 − n =2 − lim =2since < 1 ⇒ lim =0. n→∞ 3 n→∞ 3 3 n→∞ 3 n n (−2) 2 2 (b)n→∞m 3n = n→∞ −3 =0since −3 < 1. n n (c) (−4) = − 4 diverges since −> 1. 3n 3 3 n3+2 n n 2 (d) The sequence diverges because = + →∞ . 3n2 3 3n x2 2x 2 n 2 (e) By L’opital’s rule, lim= lim = lim =0 ⇒ lim =0. x→∞ ex x→∞ ex x→∞ e n→∞ en [4] 2. Find the sum of the series. ∞ ∞ 4 · 2+1 2 (a) n (b) n+1 n=0 3 n=13 Solution: ∞ n+1 ∞ n 4 · 2 2 8 (a) n = 8 = 2=24. n=0 3 n=0 3 1 − 3 ∞ 2 ∞ 2 1 n 2 1 1 (b) = = 3 = . 3n+1 3 3 31 − 1 3 n=1 n=1 3 [9] 3. Dete∞mine whether the ∞eries converges or ∞iverges. Justify your answer. 2 3n +2 1 (a) 3 (b) √ (c) 3/2 2 n=3n[ln(n)] n=0 4n +1 n=2n [ln(n)] Solution:
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