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!

Testing a series for convergence

  1. Apr 22, 2009 #1

    a.s

    User Avatar

    EDIT: The latex doesn't seem to be working at all... not exactly sure why this is. I can't delete the post, so, uh... never mind, I guess?
    EDIT v. 2.0: Yeah, so copying and pasting images from LatexIt is apparently beyond me... Thanks for the help, razored! Right, so, any help anyone can give will be well appreciated!

    1. The problem statement, all variables and given/known data
    Determine whether the following series converges or diverges. If possible, determine the sum of the series exactly. Justify your answer with the proper series test.
    [tex]\sum_{n=2}^\infty{\sqrt{n^3+3}-\sqrt{n^3-3}}[/tex]

    2. Relevant equations
    Comparison test, integral test, root test, ratio test, etc.


    3. The attempt at a solution
    Multiplying by the conjugate
    [tex]\sqrt{n^3+3}+\sqrt{n^3-3}[/tex]
    produces the series
    [tex]\sum_{n=2}^\infty{\frac{6}{\sqrt{n^3+3}+\sqrt{n^3-3}}}[/tex].

    My first inclination was to try and find a series which is clearly larger, but I'm having trouble doing that. In previous problems like this that I've seen, there's been an n in the denominator, allowing me to eliminate the terms with square roots and produce a series which is always greater.

    In this case, though, I can't, so I gave the ratio test a try, with painful results:
    [tex]\lim_{n\rightarrow\infty}\frac{\sqrt{(n+1)^3+3}+\sqrt{(n+1)^3-3}}{\sqrt{n^3+3}+\sqrt{n^3-3}}[/tex].
    I tried expanding the cubic terms, making it a bit ridiculous.
    [tex]\lim_{n\rightarrow\infty}\frac{\sqrt{n^3+3n^2+3n+4}+\sqrt{n^3+3n^2+3n-2}}{\sqrt{n^3+3}+\sqrt{n^3-3}}[/tex]
     
    Last edited: Apr 22, 2009
  2. jcsd
  3. Apr 22, 2009 #2
    Your Work :

    http://texify.com/img/%5CLARGE%5C%21%5Ctext%7BWork%20from%20poster%3A%20%7D%20%5C%5C%5Csum_%7Bn%3D2%7D%5E%5Cinfty%7B%5Csqrt%7Bn%5E3%2B3%7D-%5Csqrt%7Bn%5E3-3%7D%7D%20%5C%5C%5Csqrt%7Bn%5E3%2B3%7D%2B%5Csqrt%7Bn%5E3-3%7D%20%5C%5C%5Csum_%7Bn%3D2%7D%5E%5Cinfty%7B%5Cfrac%7B6%7D%7B%5Csqrt%7Bn%5E3%2B3%7D%2B%5Csqrt%7Bn%5E3-3%7D%7D%7D%5C%5C%5Clim_%7Bn%5Crightarrow%5Cinfty%7D%5Cfrac%7B%5Csqrt%7B%28n%2B1%29%5E3%2B3%7D%2B%5Cs%20%20qrt%7B%28n%2B1%29%5E3-3%7D%7D%7B%5Csqrt%7Bn%5E3%2B3%7D%2B%5Csqrt%7Bn%5E3-3%7D%7D%5C%5C%20%5Clim_%7Bn%5Crightarrow%5Cinfty%7D%5Cfrac%7B%5Csqrt%7Bn%5E3%2B3n%5E2%2B3n%2B4%20%20%7D%2B%5Csqrt%7Bn%5E3%2B3n%5E2%2B3n-2%7D%7D%7B%5Csqrt%7Bn%5E3%2B3%7D%2B%5Csqrt%7Bn%5E3-3%7D%7D.gif [Broken]
     
    Last edited by a moderator: May 4, 2017
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Testing a series for convergence
Loading...