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Homework Help: Comparison Test

  1. Jun 4, 2010 #1
    frustration wall.jpg

    I have hard time with latex so I attached all relevant info in that picture file.:redface:
    Thanks you in advance!
    Last edited: Jun 4, 2010
  2. jcsd
  3. Jun 4, 2010 #2
    Do not make it complicated

    take the limit(as n goes to infinity) of both sides of the inequality in the question (the inequality which has the nth rooth of the absolute value of an)

    the left side will be simply the limit which is in the root test and the right side will be 1
    so the limit of the nth root of the absolute value of an is less than 1
    so the series converges by the nth root test.
  4. Jun 4, 2010 #3
    I'm not sure if I get you right.
    The root test tell us something only if the limit is not 1, but in our case the limit is 1.
  5. Jun 4, 2010 #4
    Take the limit of both sides as I said, the limit of the right side will be 1
    so the limit of the right side (which the nth root of the absolute value of an) is LESS THAN 1 ---> The series converges by the root test.
  6. Jun 4, 2010 #5
    Sorry, but I still can't get the idea.

    You mean this expression? frustration wall.jpg How you conclude that it is less then 1?
  7. Jun 4, 2010 #6
    The inequality itself

    take the limit of both sides

    it will something like
    the lim of the left side LESS THAN the lim of the right side
  8. Jun 4, 2010 #7
    Where in the inequality you see "LESS THAN"? (It's less OR EQUAL to)
    OBVIOUSLY, the root test CAN'T be used here.
    Last edited: Jun 4, 2010
  9. Jun 4, 2010 #8
    yeah, its less than or equal
    Wait, I will do it in a picture.
  10. Jun 4, 2010 #9
    Its in the attachment.

    Attached Files:

  11. Jun 4, 2010 #10
  12. Jun 4, 2010 #11
    your website said: if L < 1 then the series absolutely convergent (hence convergent).
    Its agree with my conclusion :)
  13. Jun 4, 2010 #12
    If you mean less than or EQUAL , then you could consider the inequality starts from n=2, and this will be remove the "EQUAL" sign from the inequality
    Finally, you know that the finite sum does not affect the convergence of the series, so it will include the 1.
  14. Jun 4, 2010 #13
    The limit is LESS OR EQUAL to 1, means the limit CAN be 1, which means you CAN'T use the root test!

    What?! (we are talking about LIMIT)
  15. Jun 4, 2010 #14
    If f(n)<g(n) for n>a


    f(n)<g(n) for n>a+1
  16. Jun 4, 2010 #15
    Thank you for your help, but I'll keep looking for other ideas (since I believe your idea is wrong).
  17. Jun 4, 2010 #16
    I'm not sure if this is right but what I did was:

    let bn = 1 - 1/n^a, for 0 < a < 1, and an = bn ^ n; by virtue of the root test, this converges since the limit is between but not equal to 0 and 1.
  18. Jun 4, 2010 #17
    I'm not sure but I think it is not. Again root test can't be used here.(despite the resemblance)
  19. Jun 4, 2010 #18


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    Homework Helper

    Good for you for resisting all of the bad root test advice. You could try to apply an integral test to exp(-n^(1-a)). It gives you an incomplete gamma function, if you know how to deal with that. I mostly know this because I typed it into Maxima. It diverges as a->1, of course, but seems to be finite on a in (0,1).
  20. Jun 5, 2010 #19
    Thank you for your response.

    I'm not too familiar with the gamma function and how to deal with it,
    nevertheless I tried applying integral test to exp(-n^(1-a)), ( too much work=) and not sure how to integrate it anyway).
    Do you think there is no better way, and I should make it with the integral test?
    Last edited: Jun 5, 2010
  21. Jun 5, 2010 #20
    Intuition suggests getting from exp(-n^(1-a)) to geometric series, but I can't figure out algebra recipe to get there... =(
    Any ideas?
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