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How to compute 'special' sum

  1. Mar 4, 2009 #1
    Hello! I am trying to find a way to compute the following in an Excel spreadsheet without the use of a macro:

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

    Can I manipulate this into some form that would make it easier to put in Excel? I'm pretty stuck and any suggestions would be much appreciated!

    Thanks,

    Kerry

    [EDIT] Just wanted to make it clearer - My goal is to have some formula in one cell that references just one other cell to get the result.
     
  2. jcsd
  3. Mar 4, 2009 #2
    Unfortunately, I do not believe a closed-form solution exists for the summation

    S = 1/1^2 + 1/2^2 + ... + 1/n^2 = Sum of 1/k^2 from 1 to n.

    The whole thing converges to 1.644934066848226436472415166646, or pi^2 / 6, as n goes to infinity.

    What sort of accuracy do you need? You can always plot a range of it, and then fit a curve to it to make it look "close".
     
  4. Mar 4, 2009 #3
    Ahh, I was afraid of that. I'll just write the macro.

    Thanks for your help!

    -Kerry
     
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