几个欧拉和（Euler sum）的求解

\begin{align}&\sum\limits_{n = 1}^\infty  {\frac{1}{{{n^2}}}\left( {\sum\limits_{k = 1}^n {\frac{{{{\left( { - 1} \right)}^{k - 1}}}}{k}} } \right)\left( {\sum\limits_{k = 1}^n {\frac{{{{\left( { - 1} \right)}^{k - 1}}}}{{{k^2}}}} } \right)}; \\&\sum\limits_{n = 1}^\infty  {\frac{{{{\left( { - 1} \right)}^{k - 1}}}}{{{n^2}}}\left( {\sum\limits_{k = 1}^n {\frac{{{{\left( { - 1} \right)}^{k - 1}}}}{k}} } \right)\left( {\sum\limits_{k = 1}^n {\frac{1}{{{k^2}}}} } \right)}; \\&\sum\limits_{n = 1}^\infty  {\frac{{\left( {\sum\limits_{n = 1}^\infty  {\frac{1}{{{n^2}}}} } \right)\left( {\sum\limits_{n = 1}^\infty  {\frac{1}{{{n^3}}}} } \right)}}{{{n^2}}}}; \\&\sum\limits_{n = 1}^\infty  {\frac{1}{{{n^2}{H_n}}}} ,{H_n} = \sum\limits_{k = 1}^n  {\frac{1}{n}}. \end{align}