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Homework Help: Polynomial proof help

  1. Sep 19, 2005 #1
    "Prove that

    [tex]1^k+2^k+...+n^k[/tex]

    can be written as a polynomial in [tex]n[/tex] of degree at most [tex]k+1[/tex]."

    Isn't this kinda trivial? I mean I know the "book" solution is to prove by induction, etc, but assuming that I have the above expression, I can prove or disprove it, depending on how I interpret the question.

    If it means that it can be written in the above conditions AND NOTHING ELSE, I can easily produce a counterexample:

    [tex]1+2+3 = 3^3 - 7\times3[/tex]

    If it means that it can be written in the above conditions, but does not prohibit the existence of other solutions, then it's trivial, because the above expression can be written as

    [tex]an^{k+1}[/tex] for some real number [tex]a[/tex]
     
  2. jcsd
  3. Sep 19, 2005 #2

    shmoe

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    It doesn't mean "and nothing else", it's already written in a form that you wouldn't call a polynomial.

    No choice of a here will hold for all n. Your polynomial is supposed to equal that expression for all values of n.
     
  4. Sep 20, 2005 #3
    True, no choice of a will hold for every n, but there exists one for every n.
     
  5. Sep 20, 2005 #4

    shmoe

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    It's not a polynomial if the coefficients aren't constant. Your a will depend on n in some unspecified way, and you haven't solved the problem.
     
  6. Sep 20, 2005 #5
    No easy way out then. Damn.
     
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