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!

What convergence/divergence test can I use on this series? - Calculus II

  1. Mar 2, 2006 #1
    What test can I use on the following series in order to determine
    if it converges or diverges? Looking at it graphically it appears to
    diverge but I cannot show it analytically.

    \sum\limits_{n = 1}^\infty {\frac{{n!}}{{3n! - 1}}}

    Using the Ratio Test, here is I got thus far

    \left| {\frac{{a_{n + 1} }}{{a_n }}} \right| = \left| {\frac{{\left( {\frac{{(n + 1)!}}{{3(n + 1)! - 1}}} \right)}}{{\left( {\frac{{n!}}{{3n! - 1}}} \right)}}} \right|

    = \frac{{(n + 1)!}}{{3(n + 1)! - 1}} \cdot \frac{{3n! - 1}}{{n!}}

    = \frac{{(n + 1)n!}}{{3(n + 1)n! - 1}} \cdot \frac{{3n! - 1}}{{n!}}

    = \frac{{(n + 1)(3n! - 1)}}{{3(n + 1)n! - 1}}

    At this point, I don't see how the Ratio test will work. What test can I use on this series?

    Should I start off with long division to reduce the improper rational

    http://img164.imageshack.us/img164/4656/longdiv7nr.jpg [Broken]

    So the equivalent sum would be

    \sum\limits_{n = 1}^\infty {\left( {\frac{1}{3} + \frac{1}{{3(3n! - 1)}}} \right)}

    Which must diverge because of the 1/3 always being summed from
    1 to infinity.

    Is that the correct way to do it? If so, is it the only way? Am I
    missing a simple test that could be used?
    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Mar 3, 2006 #2
    oops nevermind I looked again and it seems u already did that.
  4. Mar 3, 2006 #3


    User Avatar
    Homework Helper

    Maybe comparison test. But the work have looks good to me.
  5. Mar 3, 2006 #4


    User Avatar
    Science Advisor

    I would be inclined to use the fact that is an does not go to 0 then [itex]\Sum a_n[/itex] does not converge!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook