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: Limit of series

  1. Feb 14, 2006 #1
    for infinity sum n=1 (2n+1) / (5n+1)

    how to find the converges or diverges??
    is it suitable to use basic comparison test??pls show me
     
  2. jcsd
  3. Feb 14, 2006 #2

    TD

    User Avatar
    Homework Helper

    If we split the fraction, we get

    [tex]\frac{{2n + 1}}{{5n + 1}} = \frac{2}{5} + \frac{3}{{25n + 5}}[/tex]

    As you can see, the second term will go to 0 if n tends to infinity but there's a remaining term 2/5. In other words: the limit of the associated sequence isn't 0 (for n going to infinity) and this is a necessary (though not sufficient) condition for the convergence of the series.
     
  4. Feb 14, 2006 #3

    HallsofIvy

    User Avatar
    Science Advisor

    Or: divide both numerator and denominator by n:
    [tex]\frac{2n+1}{5n+1}= \frac{2+\frac{1}{n}}{5+\frac{1}{n}}[/tex]

    Now what does 1/n go to as n goes to infinity?
     
  5. Feb 14, 2006 #4
    but the question ask for diverges or converges??and the answer is diverges
     
  6. Feb 14, 2006 #5

    TD

    User Avatar
    Homework Helper

    When a series doesn't converge, it diverges.
     
  7. Feb 14, 2006 #6
    is it necessary to use any rules such as the one i mentioned??or just to perform the steps showed above??
     
  8. Feb 14, 2006 #7

    TD

    User Avatar
    Homework Helper

    I don't think those are necessary, if the limit of the (positive) sequence doesn't go to 0 (which is easy to check without using those other tests), then the (positive) series doesn't converge.
     
  9. Feb 14, 2006 #8
    teng the n-th term test for divergence is an easy check for this one.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook