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Unformatted text preview: ly the improper integral 1∞ xe−x dx converges.
∞
−x
dx = 2/e < ∞ (integrate by parts and use L’Hopital’s rule) so series
1 xe
converges.
3.3 Determine if the following series converge or diverge. Give your reasoning
using complete sentences.
a)
b) ∞ ln n
2
n=1 n
∞ n!
n=1 (n + 2)! MTH 142 Exam 3 Spr 2011 Practice Problem Solutions 2 a) Series converges if and only the improper integral 1∞ ln2x dx converges.
x
Integrate by parts and use L’Hopital’s rule to see that this integral converges
to 1 so series converges. Alternately, ln x < x1/2 when x is large since by
ln
L’Hopital’s rule limx→∞ x1/x = 0, so ln2x ≤ x31/2 so 1∞ ln2x dx is convergent by
2
x
x
comparison with the integral in problem 2a) above
1
1
n!
b) (n+2)! = (n+2)(n+1) < n2 so given series converges by comparison with
pseries with p = 2 which is convergent. 3.4 For each of the following items a) and b) choose a correct conclusion
and reason from among the choices (R), (C), (I) below and provide supporting
computation. For example, if you choose (R) calcul...
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 Fall '08
 POOLE
 Calculus

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