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Stuck with L' Hospital's Rule

  1. Apr 13, 2012 #1
    1. The problem statement, all variables and given/known data
    Ʃ n/e^n converge or diverge


    2. Relevant equations



    3. The attempt at a solution

    I got this down to an improper integral using the integral test but I am weak at L'Hospitals rules and I was wondering if someone could help me out

    I have

    [itex]\int n/e^n[/itex] from 1 to infinity

    down to

    Limit b to infinity ne^n - e^n |from 1 to b

    this gives me infinity - infinity so time for L'Hospitals ( forgive my spelling)
    I know to divide by the recripical of either one but I get stuck from there
     
  2. jcsd
  3. Apr 13, 2012 #2

    Dick

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    It's probably easier to use the ratio test. If you have to use the integral test you didn't get the integral quite right. There's a sign problem (which isn't terribly important) and don't you mean e^(-n) in the integral (which is terribly important)?
     
  4. Apr 13, 2012 #3
    ok i will try the ratio test for this... I am studying for a test and have already worked this problem and turned it in when I got it back graded the only comments were that it was infinity-infinity and needs L'Hospitals rule. I had originally put infinity - infinity so it diverges. The original problem is correct it is n / e^n not e ^-n
     
  5. Apr 13, 2012 #4

    Dick

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    No, I meant your integral should have been -n/e^n - 1/e^n or -ne^(-n) - e^(-n). It's not infinity-infinity.
     
  6. Apr 13, 2012 #5
    Ok I did the Ratio test could you please check my work?

    [itex]\frac{n}{e^n}[/itex]

    [itex]\frac{n+1}{e^(n+1)}[/itex] * [itex]\frac{e^n}{n}[/itex]

    so all e's cancel except 1 giving me

    [itex]\frac{n+1}{e*n}[/itex] the limit of this is ∞/∞ with L'Hopitals i have

    [itex]\frac{1}{e}[/itex] which is less then 1 so ratio test says converges
     
    Last edited: Apr 13, 2012
  7. Apr 13, 2012 #6

    Dick

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    Yes, the limit of the ratio is 1/e so it converges.
     
  8. Apr 13, 2012 #7
    Awesome thanks... and I will try to work on form... that was the edited version you should have seen what i had first lol!
     
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