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'value of Dirichlet eta function at $s = 2$'
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| Title of object: |
value of Dirichlet eta function at $s = 2$ |
| Canonical Name: |
ValueOfDirichletEtaFunctionAtS2 |
| Type: |
Result |
| Created on: |
2008-09-08 16:57:56 |
| Modified on: |
2008-09-08 16:57:56 |
| Classification: |
msc:11M41 |
Revision comment (for changes between this and next version):
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Content:
The value
$$\eta(2) = 1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\!-\ldots$$
of the Dirichlet eta function can be found by using the Fourier cosine series \PMlinkescapetext{expansion} of the function \,$x \mapsto x\!-\!x^2$
on the interval \,$[0,\,1]$:
\begin{align}
x\!-\!x^2 \;=\; \frac{1}{6}-\frac{1}{\pi^2}\sum_{n=1}^\infty\frac{\cos{2n\pi x}}{n^2}
\quad \mbox{for}\;\; 0 \leqq x \leqq 1
\end{align}
Substituting\, $x := \frac{1}{2}$\, to the equation (1) yields
$$\frac{1}{4} \;=\; \frac{1}{6}-\frac{1}{\pi^2}\sum_{n=1}^\infty\frac{\cos{n\pi}}{n^2}
\;=\; \frac{1}{6}+\frac{1}{\pi^2}\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^2},$$
which we can solve to the form
$$\zeta(2) \;=\; \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^2} \;=\; \frac{\pi^2}{12}.$$
This result could be obtained very simply by using the functional equation connecting Dirichlet eta function to Riemann zeta function.
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