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Convergence Tests

  1. Jun 9, 2009 #1
    1. The problem statement, all variables and given/known data

    Here is a worked example:

    http://img79.imageshack.us/img79/1909/88760168.gif [Broken]

    I don't understand why they used the "limit comparison test" and not the "comparison test".

    Why can't we just conclude that the series [tex]\sum^{\infty}_{n=1} (\frac{n+1}{n^2 +1})^3[/tex] converges by comparison with [tex]\frac{1}{n^3}[/tex]?

    What's the necessity of using the limit comparison test?

    Furthermore, I would like to know in which cases it is more suitable to use the comparison test and when to use the limit comparison test. I appreciate some guidance here. Thanks.
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Jun 9, 2009 #2

    Mark44

    Staff: Mentor

    To use the comparison test you would have to show that ((n + 1)/(n^2 + 1))^3 <= 1/n^3, for all n larger than some value N_0. It might not be very easy to establish this inequality. The limit comparison gets around this problem.
     
  4. Jun 9, 2009 #3
    How did you first inspect that it would not be very easy to establish that [tex](\frac{n+1}{n^2 +1})^3 \leq \frac{1}{n^3}[/tex]? :smile:
     
  5. Jun 9, 2009 #4
    If I were using the regular comparison test on this, I would say

    [tex]n+1\le n+n= 2n[/tex]

    and

    [tex]n^2+1\ge n^2[/tex]

    so that

    [tex]\frac{n+1}{n^2+1}\le\frac{2n}{n^2}=\frac{2}{n}[/tex]

    and thus

    [tex]\left(\frac{n+1}{n^2+1}\right)^3\le\frac{8}{n^3}[/tex]
     
  6. Jun 10, 2009 #5

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Without context I assume the purpose of the example is to demonstrate the limit comparison test, not demonstrate how to prove the convergence of that series
     
  7. Jun 11, 2009 #6
    Hi!
    I can't follow exactly what you've done here, could you please explain a little more?
     
  8. Jun 12, 2009 #7

    Mark44

    Staff: Mentor

    If you want to show that, for example, a/b < c/d, the best way to do this is to show that a < c and b > d.

    You want the numerator on the left to be smaller than the one on the right, and you want the denominator on the left to be bigger than the one on the right. A larger denominator is associated with a smaller value.

    I assume that that was the part of Billy Bob's work that you didn't understand.
     
  9. Jun 12, 2009 #8
    I get it now, thank you.
     
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