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Determine whether the following series converges

  1. Nov 5, 2006 #1
    Could someone plz explain how they got from step 1 to step 2, i don't know what hap'd to the factorial part :S

    Question: Determine whether the following series converges.

    n= infinity
    Sum ln (n)
    n=1 n!

    Using the Ratio Test: lim absolute ((An+1)/(An))

    Step 1. => lim (ln(n+1)/(n+1)!) x (n!/(ln (n))

    Step 2. => lim ln(n+1)/((n+1)ln(n))

    Link to solution: http://www.maths.uq.edu.au/courses/MATH1051/Semester2/Tutorials/prob9sol.pdf [Broken]

    It's question 1. vi)
    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Nov 5, 2006 #2
    n!/(n + 1)! = 1/(n + 1).

    If you write out the factorials for n = 1, 2, 3 you should see why this is the case.
  4. Nov 14, 2006 #3
    Using this equality e^x >= x + 1 > x for all x, we have e^n > n for all n. Thus, ln(n) < n for all n. This implies 0 < ln(n)/n! < 1/(n-1)! for all n>1
    The series (Sum 1/(n-1)!) is convergent, so is the series (Sum ln(n)/n!)
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