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Inequality question

  1. Mar 19, 2005 #1
    Hello everyone, could someone please help me out with the following question?

    Q. Prove that the following inequality holds for all natural numbers n and rewrite the inequality using summation notation.

    [tex]1^3 + 2^3 + ... + \left( {n - 1} \right)^3 < \frac{1}{4}n^4 < 1^3 + 2^3 + ... + n^3 [/tex]

    For the summation notation part I got: [tex]\sum\limits_{k = 1}^n {\left( {k - 1} \right)^3 } < \frac{1}{4}n^4 < \sum\limits_{k = 1}^n {k^3 } [/tex]

    I managed to prove that the inequality is true. However, there seems to be some sort of symmetry in the inequality. Is there a 'shortcut' to this question or does it need to be done the way that I did? That is, showing separately that the left and right parts of the inequality are true and then combining the results. I'd like to know partly because I want to enhance my understanding of questions of this type. Any ehlp appreciated.
     
  2. jcsd
  3. Mar 19, 2005 #2
    If you want to approach it analytically:
    Notice that the left-hand side is a lower sum and the right-hand side is an upper sum of the integral
    [tex]\int_0^{n}x^3 dx[/tex]

    This directly yields:

    [tex]1^3 + 2^3 + ... + \left( {n - 1} \right)^3 \leq \frac{1}{4}n^4 \leq 1^3 + 2^3 + ... + n^3 [/tex]

    It is not difficult to show that equality is not possible.
     
  4. Mar 19, 2005 #3
    Thank you for your help Timbuqtu.
     
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