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Another question about Pi

  1. Nov 15, 2005 #1
    I saw the other thread, but figured this question was sufficiently distinct to warrent a new thread
    I was recently looking at this series
    [tex]\[
    \sum_{n=1}^\infty \frac{1}{n^{2}} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots + \frac{1}{n^{2}} = \frac{\pi^{2}}{6} \approx 1.645
    \][/tex]
    My math teacher gave me the answer of [tex]$\pi^{2}$[/tex]/6, and by looking at the sum numerically it seems to come up that way. I'm wondering, though. Where does the [tex]$\pi$[/tex] come from?
    I tried to find an expression for the k'th term of the sum, and came up with this
    [tex]\begin{center}
    \begin{tabular}{| l | c | }
    \hline
    $k$ & $S_k$ \\ \hline
    1 & 1 \\ \hline
    2 & $\frac{5}{4}$ \\ \hline
    3 & $\frac{49}{36}$ \\ \hline
    4 & $\frac{820}{576}$ \\ \hline
    5 & $\frac{21076}{14400}$ \\ \hline
    6 & $\frac{773136}{518400}$ \\ \hline
    \end{tabular}
    \end{center}[/tex]
    Note that the fractions are all left unsimplified. I noticed that all of the denominators were perfect squares:
    [tex]\begin{center}
    \begin{tabular}{| l | c | }
    \hline
    k & S_k \\ \hline
    1 & 1^2 \\ \hline
    2 & 2^2 \\ \hline
    3 & 6^2 \\ \hline
    4 & 24^2 \\ \hline
    5 & 120^2 \\ \hline
    6 & 720^2 \\ \hline
    \end{tabular}
    \end{center}[/tex]
    As you can see, the denominator of the fraction works out to be k!^2. However, I still can't figure out where the pi comes from, or, for that matter, see any pattern in the numerator. Any ideas?
     
    Last edited: Nov 15, 2005
  2. jcsd
  3. Nov 15, 2005 #2
  4. Nov 15, 2005 #3

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    None of the partial sums will have a pi in them, only approximations (the partial sums are all rational).

    This has many ways to prove it:

    http://www.maths.ex.ac.uk/~rjc/etc/zeta2.pdf

    Depending on what you know, you might find Euler's orignal method (#7 in the above) the easiest to folow. More details on this method can be found in (eq (20) and on):

    http://plus.maths.org/issue19/features/infseries/

    Though Euler hadn't actually justified his product form for sin(x), it can be done.
     
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