1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Does SUM log(1+1/n)/n converge?

  1. Mar 15, 2006 #1
    Prove that [tex]\sum \frac{1}{n}\log(1+\frac{1}{n})[/tex] converges.

    I tried a batch of tests, but none works. Last time Raabe's test was give to me upside down. I wonder if any other tests are give to me wrongly as well.
  2. jcsd
  3. Mar 15, 2006 #2
    Have you tried comparing it to 1/n^2?
  4. Mar 15, 2006 #3
    I can't get it to work out
  5. Mar 15, 2006 #4


    User Avatar
    Science Advisor
    Homework Helper

    Post the details of your comparison...

    What bounds do you know for log(1+x)?
  6. Mar 15, 2006 #5
    It's unbounded.

    I figured if [tex]\exp(\frac{1}{n}\log(1+\frac{1}{n}))=\exp(\frac{1}{n})+\exp(\frac{1}{n})/n \rightarrow 1[/tex] as [tex]n\rightarrow\infty[/tex], so at least the terms go to zero, although it's insufficient.

    I could use the Mercator series, but we haven't seen it yet.
    Last edited: Mar 15, 2006
  7. Mar 16, 2006 #6


    User Avatar
    Science Advisor

    The first thing that was suggested was comparison- have you tried that?

    The Taylor's series for ex is 1+ x + (1/2)x2+ ... so if x is positive, ex> 1+ x and x> log(1+ x) for all x. In particular, 1/n> log(1+ 1/n) for all n.
  8. Mar 16, 2006 #7


    User Avatar
    Science Advisor
    Homework Helper

    It's bounded by a constant in the region you're concerned with. You are looking at log(1+x) where x=1/n, so 1<1+x<2.

    A constant bound will not suffice here though, you need some indication of how fast this term is going to zero. Try the log(1+x)<x Halls mentioned, this was the one I was fishing for (one function can "bound" another).
  9. Mar 16, 2006 #8
    I got it. Thanks for the help.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook