Results 1 to 4 of 4

Math Help - Prove this sum tends to zero

  1. #1
    Junior Member
    Joined
    Oct 2009
    Posts
    36

    Prove this sum tends to zero

    Show \sum_{i=2}^{n} \frac{i}{n^2 \log i} \rightarrow 0 as n \rightarrow \infty.

    I can solve a problem if I can prove this sum tends to zero as n tends to infinity but I'm struggling with the last stage. I'm not sure if I'm missing something obvious...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,407
    Thanks
    1294
    Quote Originally Posted by Boysilver View Post
    Show \sum_{i=2}^{n} \frac{i}{n^2 \log i} \rightarrow 0 as n \rightarrow \infty.

    I can solve a problem if I can prove this sum tends to zero as n tends to infinity but I'm struggling with the last stage. I'm not sure if I'm missing something obvious...
    Can you evaluate

    \sum_{i = 2}^{\infty} \frac{i}{n^2 \log{i}}?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by Prove It View Post
    Can you evaluate

    \sum_{i = 2}^{\infty} \frac{i}{n^2 \log{i}}?

    I can't see how that helps, not to mention that that infinite series diverges so he won't be able to evaluate it, nor anyone else will.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Aug 2008
    Posts
    903
    How about writing it as:

    \lim_{n\to\infty}\left\{\frac{\displaystyle{\sum_{  k=2}^n \frac{k}{\log(k)}}}{n^2}\right\}

    The general expression for the sum is \frac{2}{\log(2)}+\frac{3}{\log(3)}+\cdots+\frac{n  }{\log(n)}

    Isn't each of those less than \frac{n}{\log(n)} and you have less than n of them then what does that say about the limit?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove a/b and a/c then a/ (3b-7c)
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 23rd 2010, 05:20 PM
  2. prove,,,
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 1st 2010, 09:02 AM
  3. Prove |w + z| <= |w| +|z|
    Posted in the Algebra Forum
    Replies: 3
    Last Post: February 28th 2010, 05:44 AM
  4. Replies: 2
    Last Post: August 28th 2009, 02:59 AM
  5. How to prove that n^2 + n + 2 is even??
    Posted in the Algebra Forum
    Replies: 3
    Last Post: November 30th 2008, 01:24 PM

Search Tags


/mathhelpforum @mathhelpforum