Unformatted text preview: ly the improper integral 1∞ xe−x dx converges.
dx = 2/e < ∞ (integrate by parts and use L’Hopital’s rule) so series
3.3 Determine if the following series converge or diverge. Give your reasoning
using complete sentences.
b) ∞ ln 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.
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
L’Hopital’s rule limx→∞ x1/x = 0, so ln2x ≤ x31/2 so 1∞ ln2x dx is convergent by
comparison with the integral in problem 2a) above
b) (n+2)! = (n+2)(n+1) < n2 so given series converges by comparison with
p-series 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...
View Full Document
- Fall '08
- Calculus, Cos, Mathematical Series, dx, dθ