Results 1 to 7 of 7

Math Help - limit of fraction with summation

  1. #1
    Newbie
    Joined
    Aug 2012
    From
    Czech
    Posts
    2

    Unhappy limit of fraction with summation

    hi,
    it has been some time since i last saw calculus and I'm stuck at one excercise. The point is just to find limit of

     \frac{ln(x)}{\sum_{k=1}^n \frac{1}{k}}

    What I would do is just solving the summation as definite integral and then differentiating the fraction via l'Hospital rule. Is that an ok way to go?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Aug 2012
    From
    sanaa
    Posts
    18
    Thanks
    3

    Re: limit of fraction with summation

    But the summation is in terms of k ,, and the numerator is in term of x !
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,383
    Thanks
    1474
    Awards
    1

    Re: limit of fraction with summation

    Quote Originally Posted by pyjong View Post
    The point is just to find limit of
     \frac{ln(x)}{\sum_{k=1}^n \frac{1}{k}}
    What I would do is just solving the summation as definite integral and then differentiating the fraction via l'Hospital rule. Is that an ok way to go?
    This question does not make sense. Where is the limit? x\to? or n\to\infty~?
    What is the exact wording of the question?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Aug 2012
    From
    Czech
    Posts
    2

    Re: limit of fraction with summation

    oh .. sorry my bad, there is supposed to be n instead of x and the limit is in +inf
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    10,964
    Thanks
    1008

    Re: limit of fraction with summation

    Quote Originally Posted by pyjong View Post
    oh .. sorry my bad, there is supposed to be n instead of x and the limit is in +inf
    It is well known that the Harmonic series is divergent, so the denominator tends to \displaystyle \begin{align*} \infty \end{align*}, and the numerator also tends to \displaystyle \begin{align*} \infty \end{align*}. Since this goes to \displaystyle \begin{align*} \frac{\infty}{\infty} \end{align*}, you should be able to apply L'Hospital's Rule.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,383
    Thanks
    1474
    Awards
    1

    Re: limit of fraction with summation

    Quote Originally Posted by pyjong View Post
    oh .. sorry my bad, there is supposed to be n instead of x and the limit is in +inf
    So it is \lim _{n \to \infty }}\frac{{\ln (n)}}{{\sum\limits_{k = 1}^n {\frac{1}{k}} }}
    For the harmonic series let {H_n} = \sum\limits_{k = 1}^n {\frac{1}{k}} .
    Last edited by Plato; August 25th 2012 at 09:33 AM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Jul 2010
    From
    Vancouver
    Posts
    432
    Thanks
    16

    Re: limit of fraction with summation

    There is another way to do it but it requires a piece of knowledge. The Euler-Mascheroni constant is defined as

    \gamma = \lim_{n\to \infty} (H_n - \ln n)

    where H_n is the n-th Harmonic number. Now we have

    \lim_{n\to \infty} \frac{\ln n}{H_n} =\lim_{n\to \infty} \frac{\ln n - H_n + H_n}{H_n} = \lim_{n\to \infty} \frac{\frac{\ln n - H_n}{H_n} + 1}{1}

    What's the next step? If you want to see, click "show"
    Spoiler:
    Now, as n\to \infty the numerator in the inner fraction tends to -\gamma and the denominator tends to +\infty. Therefore, the limit is

    \lim_{n\to \infty} \frac{\ln n}{H_n} = 1
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Summation of fraction
    Posted in the Algebra Forum
    Replies: 4
    Last Post: April 17th 2011, 10:53 PM
  2. limit of functions in a fraction
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: April 24th 2010, 12:48 PM
  3. limit and summation help
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 21st 2009, 04:44 AM
  4. Limit fraction.
    Posted in the Calculus Forum
    Replies: 5
    Last Post: May 22nd 2009, 06:18 AM
  5. Limit of a Summation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 20th 2007, 06:40 AM

Search Tags


/mathhelpforum @mathhelpforum