- #1
johnw188
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I saw the other thread, but figured this question was sufficiently distinct to warrent a new thread
I was recently looking at this series
\[<br /> \sum_{n=1}^\infty \frac{1}{n^{2}} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots + \frac{1}{n^{2}} = \frac{\pi^{2}}{6} \approx 1.645<br /> \]
My math teacher gave me the answer of $\pi^{2}$/6, and by looking at the sum numerically it seems to come up that way. I'm wondering, though. Where does the $\pi$ come from?
I tried to find an expression for the k'th term of the sum, and came up with this
\begin{center}<br /> \begin{tabular}{| l | c | }<br /> \hline<br /> $k$ & $S_k$ \\ \hline<br /> 1 & 1 \\ \hline<br /> 2 & $\frac{5}{4}$ \\ \hline<br /> 3 & $\frac{49}{36}$ \\ \hline<br /> 4 & $\frac{820}{576}$ \\ \hline<br /> 5 & $\frac{21076}{14400}$ \\ \hline<br /> 6 & $\frac{773136}{518400}$ \\ \hline<br /> \end{tabular}<br /> \end{center}
Note that the fractions are all left unsimplified. I noticed that all of the denominators were perfect squares:
\begin{center}<br /> \begin{tabular}{| l | c | }<br /> \hline<br /> k & S_k \\ \hline<br /> 1 & 1^2 \\ \hline<br /> 2 & 2^2 \\ \hline<br /> 3 & 6^2 \\ \hline<br /> 4 & 24^2 \\ \hline<br /> 5 & 120^2 \\ \hline<br /> 6 & 720^2 \\ \hline<br /> \end{tabular}<br /> \end{center}
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?
I was recently looking at this series
\[<br /> \sum_{n=1}^\infty \frac{1}{n^{2}} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots + \frac{1}{n^{2}} = \frac{\pi^{2}}{6} \approx 1.645<br /> \]
My math teacher gave me the answer of $\pi^{2}$/6, and by looking at the sum numerically it seems to come up that way. I'm wondering, though. Where does the $\pi$ come from?
I tried to find an expression for the k'th term of the sum, and came up with this
\begin{center}<br /> \begin{tabular}{| l | c | }<br /> \hline<br /> $k$ & $S_k$ \\ \hline<br /> 1 & 1 \\ \hline<br /> 2 & $\frac{5}{4}$ \\ \hline<br /> 3 & $\frac{49}{36}$ \\ \hline<br /> 4 & $\frac{820}{576}$ \\ \hline<br /> 5 & $\frac{21076}{14400}$ \\ \hline<br /> 6 & $\frac{773136}{518400}$ \\ \hline<br /> \end{tabular}<br /> \end{center}
Note that the fractions are all left unsimplified. I noticed that all of the denominators were perfect squares:
\begin{center}<br /> \begin{tabular}{| l | c | }<br /> \hline<br /> k & S_k \\ \hline<br /> 1 & 1^2 \\ \hline<br /> 2 & 2^2 \\ \hline<br /> 3 & 6^2 \\ \hline<br /> 4 & 24^2 \\ \hline<br /> 5 & 120^2 \\ \hline<br /> 6 & 720^2 \\ \hline<br /> \end{tabular}<br /> \end{center}
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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