I saw the other thread, but figured this question was sufficiently distinct to warrent a new thread(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Another question about Pi

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