Results 1 to 2 of 2

Math Help - Integral Proof,

  1. #1
    Junior Member
    Joined
    Jan 2009
    Posts
    57

    Integral Proof,

    Hi,

    I am trying to prove:

    <br />
\sum_{k=1}^{n-1} k^p < \frac{n^{p+1}}{p+1} < \sum_{k=1}^{n} k^p<br />

    I have tried a few things, my problem is that I cannot seem to find a general formula for \sum_{k=1}^{n-1} k^p

    I know that \sum_{k=1}^{n-1} k^2 = \frac{n^3}{3}-\frac{n^2}{2}+\frac{n}{6}

    But that is only order 2. I don't know if there is a general method for order p.

    Thanks
    Regards
    Craig.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by craigmain View Post
    Hi,

    I am trying to prove:

    <br />
\sum_{k=1}^{n-1} k^p < \frac{n^{p+1}}{p+1} < \sum_{k=1}^{n} k^p<br />

    I have tried a few things, my problem is that I cannot seem to find a general formula for \sum_{k=1}^{n-1} k^p

    I know that \sum_{k=1}^{n-1} k^2 = \frac{n^3}{3}-\frac{n^2}{2}+\frac{n}{6}

    But that is only order 2. I don't know if there is a general method for order p.

    Thanks
    Regards
    Craig.
    Do you know to convert summations to integrals? Consider the curve x^p. If we try to approximate the area by dividing the x-axis into pieces chosen in two different ways, you can solve the problem. Also compare your approximations to the integrals to get the inequalities..

    For starters, observe:
    \frac{n^{p+1}}{p+1} < \sum_{k=1}^{n} k^p \Leftrightarrow  \frac{1}{p+1} < \frac1{n}\sum_{k=1}^{n} \left(\frac{k}{n}\right)^p and also observe that \int_0^1 x^p \, dx = \frac{1}{p+1}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Integral proof ..
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 15th 2010, 03:25 PM
  2. Integral Proof
    Posted in the Calculus Forum
    Replies: 4
    Last Post: May 14th 2010, 03:29 AM
  3. Integral proof
    Posted in the Calculus Forum
    Replies: 3
    Last Post: June 17th 2009, 05:30 AM
  4. Help with Integral proof
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 21st 2009, 06:09 PM
  5. Replies: 6
    Last Post: August 4th 2007, 09:48 AM

Search Tags


/mathhelpforum @mathhelpforum