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Finite series

  1. Mar 30, 2012 #1
    How to prove (not by induction)

    [itex]1^{2}+2^{2}+\ldots+n^{2}=\frac{n(n+1)(2n+1)}{6}[/itex]?

    What is the general approach for similar series, say, [itex]1^{1}+2^{2}+\ldots+n^{n}[/itex]?
     
  2. jcsd
  3. Mar 31, 2012 #2
    For your first question, the sum is the solution to the difference equation S(n)-S(n-1)=n^2 subject to the initial condition S(1)=1. Since the difference is 2nd order polynomial, the solution is 3rd order polynomial, now you know how to proceed. For your second question, since the difference is n^n, no known simple function of n has such difference, therefore no simple solution.
     
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