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# math 2004 T4Solutions.pdf

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Carleton University

Mathematics

MATH 1005

Ida Toivonen

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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