1. Not finding help here? Sign up for a free 30min 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!

Series againand again

  1. Nov 9, 2004 #1
    Sorry about the title everyone but ive posted numerous threads on series and I had to choose an apropriate title :tongue2:
    The problem asks to use the ratio test, and determine for which values of x the test is conclusive-either converging or diverging. Then check those cases where the test is inconclusive by some other means.

    here is the the series [tex]\sum_{n=3}^{\infty}\frac{x^n}{n3^n}[/tex]...converge or diverge here is what i did [tex]\frac{a_{n+1}}{a_n}[/tex] and that came out to be [tex]\frac{x^{n+1}}{(n+1)(3^{n+1})}[/tex] multiplie by the [tex]\frac{n3^{n}}{x^{n}}[/tex] and after you cross out similar variables and it comes out to be

    [tex]\lim_{x\rightarrow \infty}\frac{xn}{3(n+1)}[/tex]
     
    Last edited: Nov 9, 2004
  2. jcsd
  3. Nov 9, 2004 #2

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    [tex]\frac{a_{n+1}}{a_n}=\frac{x^{n+1}n3^n}{x^n(n+1)3^{n+1}}=\frac{nx}{(n+1)3}[/tex]
     
  4. Nov 9, 2004 #3
    thanks galileo but I got that far just had problems latexing it
     
  5. Nov 9, 2004 #4
    So you get

    [tex]\lim_{n \to \infty} \left| \frac{n}{(n+1)}\frac{x}{3}\right| = \frac{|x|}{3}[/tex]

    And you know this is stricly less than 1 for it to be conclusively converging, strictly greater than 1 to be conclusively diverging, and inconclusive at 1. So you must test the values for which the expression equals one.

    --J
     
  6. Nov 10, 2004 #5

    James R

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The remaining cases are x=3 and x=-3

    If x=3 then the series reduces to the harmonic series, which diverges.

    If x=-3 then we have an alternating series, and we can use the Leibnitz test (whose exact conditions escape me right now).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Series againand again
  1. Series again (Replies: 10)

  2. Physics again (Replies: 5)

  3. HELP (again) (Replies: 5)

  4. Acceleration again (Replies: 9)

  5. Uncertainty Again (Replies: 0)

Loading...