Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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!)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook