几个逼格稍高的积分级数题
先是证明两个积分成立：
\begin{align}\int_0^{ + \infty } {\frac{{\sin nx}}{{x + \frac{1}{{x + \frac{2}{{x + \frac{3}{{x + \cdots }}}}}}}}dx} &= \frac{{\sqrt {\frac{\pi }{2}} }}{{n + \frac{1}{{n + \frac{2}{{n + \frac{3}{{n + \cdots }}}}}}}}\\\int_0^{ + \infty } {\frac{{\sin \frac{{n\pi x}}{2}}}{{x + \frac{{{1^2}}}{{x + \frac{{{2^2}}}{{x + \frac{{{3^2}}}{{x + \cdots }}}}}}}}dx} &= \frac{1}{{n + \frac{{{1^2}}}{{n + \frac{{{2^2}}}{{n + \frac{{{3^2}}}{{n + \cdots }}}}}}}}.\end{align}
接着是两个级数题，判断它们是否成立：
\begin{align}\sum\limits_{n = 0}^{ + \infty } {\left[ {\left( {1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{{2n + 1}}} \right) \cdot \frac{1}{{{5^n}\left( {2n + 1} \right)}}} \right]} &= \frac{{{\pi ^2}}}{{4\sqrt 5 }} - \frac{{\sqrt 5 }}{{24}}{\left( {\ln \left( {2 + \sqrt 5 } \right)} \right)^2}\\\sum\limits_{n = 0}^{ + \infty } {\left[ {\left( {1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{{2n + 1}}} \right) \cdot \frac{1}{{{9^n}\left( {2n + 1} \right)}}} \right]} &= \frac{{{\pi ^2}}}{8} - \frac{3}{8}{\left( {\ln 2} \right)^2}\end{align}
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