一个组合极限题目

求$\displaystyle\lim_{x \to {\pi ^ - }} \frac{{{{\left( {\int_0^{ + \infty } {\sin x\sin \sqrt x } dx} \right)}^2}\sum\limits_{n = 1}^\infty {\sum\limits_{m = 1}^\infty {{{\left( { - 1} \right)}^{m + n}}\frac{1}{{n\left( {m + 2n} \right)}}\sum\limits_{n = 1}^\infty {\frac{{{H_n}\left( {\zeta \left( 2 \right) - \sum\limits_{k = 1}^\infty {\frac{1}{{{k^2}}}} } \right)}}{n}} } } }}{{{n^2}\left( {1 - \sin \frac{\pi }{{2n}} \cdot \sum\limits_{k = 1}^{n - 1} {\sin \left( {\frac{{\left( {2[ {\sqrt {kn} }] + 1} \right)\pi }}{{2n}}} \right)} } \right)\left( {\frac{1}{{4{{\cos }^2}\frac{x}{2}}} - \sum\limits_{n = 1}^\infty {\frac{{n\cosh \left( {nx} \right)}}{{\sinh \left( {n\pi } \right)}}} - \frac{1}{{12}}} \right)}}$
其中"[$x$]"为gauss取整,$H_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$
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