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How to obtain the closed form of Euler sum

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\[\sum\limits_{n = 1}^\infty {\frac{{{\zeta _n}\left( 7 \right)}}{{{n^2}}}{{\left( { - 1} \right)}^{n - 1}}} = ?\;\;\]
\[{\zeta _n}\left( s \right) = \sum\limits_{k = 1}^n {\frac{1}{{{k^s}}}} .\]

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