关于$\pi$的级数
$$\sum\limits_{n = 0}^\infty {\frac{{1 + 14n + 76{n^2} + 168{n^3}}}{{{2^{20n}}}}{{\left( \begin{array}{c}2n\\n\end{array} \right)}^7}} = \frac{{32}}{{{\pi ^3}}}.$$
求$$\int_0^\infty {\frac{1}{r}{e^{ - \frac{r}{{{r_0}}}}}\sin krdr} = \frac{1}{{{r_0}}}\int_1^\infty {dx} \int_0^\infty {{e^{ - \frac{{rx}}{{{r_0}}}}}\sin krdr} .$$
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问题集.
1.若$\alpha\beta=\pi$,则$$\sqrt \alpha \int_0^\infty {\frac{{{e^{ - {x^2}}}dx}}{{\cosh \alpha x}}} = \sqrt \beta \int_0^\infty {\frac{{{e^{ - {x^2}}}dx}}{{\cosh \beta x}}} .$$
2.求证\[\int_{ - \infty }^\infty {\frac{{{e^{7\pi x}}}}{{{{\left( {{e^{3\pi x}} + {e^{ - 3\pi x}}} \right)}^3}\left( {1 + {x^2}} \right)}}dx} = \frac{\pi }{8} + \frac{{4\left( {837\sqrt 3 + 5\pi \left( {161 - 75\sqrt 3 \pi } \right)} \right)}}{{3375{\pi ^2}}}.\]
3.2016.12.30
\[\int_{ - \infty }^\infty {\frac{{{x^3}\sin x}}{{{x^4} + 2{x^2} + 1}}dx} = \frac{\pi }{{2e}}.\]
4.求\[\sum\limits_{k = 1}^\infty {\frac{{{{\left( { - 1} \right)}^{k - 1}}\ln \left( {2k - 1} \right)}}{{2k - 1}}} .\]
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