I always find the formula for $k\pi ^n$ when $n$ is an even number and $k$ is a rational number, but I did't find for an odd number.
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4What do you mean by a "formula" if $k\pi^n$ is not itself what you seek? – hmakholm left over Monica Oct 29 '14 at 21:59
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Maybe showing us your formula for the even case will shed some light? – Platehead Oct 29 '14 at 22:03
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For example, there is this formula $\frac{\pi ^2}{6}=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}.......$ – E.H.E Oct 29 '14 at 22:03
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$\pi^2=\pi^2+0+0+0+\cdots$ – Mariano Suárez-Álvarez Oct 29 '14 at 23:08
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The Dirichlet beta function may satisfy you : $$\beta(x):=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^x}$$ with the table of values : \begin{array} {c|c} n&\beta(n)\\ \hline 1&\frac {\pi}4\\ 2&K\\ 3&\frac {\pi^3}{32}\\ 4&\beta(4)\\ 5&\frac {5\,\pi^5}{1536}\\ \end{array} with $K$ the Catalan constant.
Here the $n$ even cases are the difficult ones as opposed to the $\zeta$ odd cases !
A parallel with $\zeta$ is proposed in this thread (Euler numbers replacing Bernoulli numbers...).
Raymond Manzoni
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I suspect the formula for $n$ even you are talking about comes from the riemann zeta function seen here. In the same articles you can read that a general formula for $\zeta(2k+1)$ is still an open problem.
AlexR
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