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Need help for three integrals

  1. Feb 17, 2012 #1
    Been a long time I had my integral class so I forgot almost everything I knew... I need to integrate to see if the serie converge (limn→∞ an = 0). Thus, there is a theorem of the integral, if you evaluate the limit of the integral of a serie when it tends to the infinite minus when x=1 you can determine if the serie is convergent or divergent...

    I can`t find a way to start the problems:

    1. limt→∞ ∫ [1, t] sin(1/x) dx
    2. limt→∞ ∫ [1, t] xe^(-2x) dx
    3. limt→∞ ∫ [1, t] 1/(1+x^(1/2)) dx

    You can also dirrectly evaluate the limit when the serie tend to infinite but I haven't found any way to match with the answers... (1. Divergent 2. Convergent 3. Divergent)

    The #1 when x tend to infinite, 1/x tend to zero thus sin(1/x) tend to zero also... The serie should be converging if so ?!?!
    The #2 should give me an integral like this one:

    1/(-2)^2 (-2x-1)e^(-2x) + C
    -1/4 * (2x+1)e^(-2x) + C

    But you can see right away if you evaluate the integral that it tends to infinite which is not converging :frown:

    For the two others I have no idea! Any help is welcome thanks!
     
    Last edited: Feb 17, 2012
  2. jcsd
  3. Feb 17, 2012 #2

    HallsofIvy

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    x is not going to infinity. It is the upper limit on the integral that is going to infinity. x ranges from 1 to infinity in that integral.
    What you get for the integral is correct but, as t goes to infinity, it does NOT go to infinity. Why would you think it does?

    For (3) let [itex]u= 1+ x^{1/2}[/itex]
     
  4. Feb 17, 2012 #3
    Just making sure I understand you correctly (I am natively speaking french).

    The serie Ʃ∞n=1 ne^(-2n) = e^-2 + 2e^(-4) + ... + ne^(-2n)

    I can clearly see here the serie is decreasing and convergent because it gets closer to a number since an > an+1.

    I would like to prove it with the integral theorem but I can't find a way to make the limt→∞ ∫ [1, t] xe^(-2x) dx = 0 thus converging.
     
  5. Feb 17, 2012 #4
    Quoting my notes:

    If ∫[1,∞[ f(x) is convergent <=> Ʃ∞n=1 an is convergent where an = f(x)

    So the goal is to prove the integral is convergent to find out if the serie is converging.
     
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