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\samepage \begin{displaymath}\sum_{k=1}^\infty \left(1 + {1 \over 2} + \cdots + {1 \over k}\right)^4 \, (k + 1)^{-5} \end{displaymath}
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{\newpage
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\samepage \begin{displaymath}= -\frac{29}{2} \zeta(9)
       + \frac{37}{2} \zeta(4) \zeta(5)\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}+ \frac{33}{4} \zeta(3) \zeta(6) - \frac{8}{3} \zeta^3(3)
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{\newpage
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\samepage \begin{displaymath}\sqrt{\pi}  \doteq \frac{1}{10^5} \sum_{n=-\infty}^{\infty} e^{-n^2/10^{10}}\end{displaymath}
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{\newpage
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\samepage \begin{displaymath}a_{n+1}:=\frac{a_n+b_n}{2}, \quad
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{\newpage
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\samepage \begin{displaymath}\lim_{n\rightarrow \infty} a_n =
 \lim_{n\rightarrow \infty} b_n =  
 \frac{\pi/2}{ \int_{0}^1
  \frac{dt}{\sqrt{1-t^4}}} \end{displaymath}
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{\newpage
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\samepage \begin{displaymath}=1.1981402347355922075...\end{displaymath}
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\samepage \begin{displaymath}({{e^{\pi \,\sqrt {163}}}-744})^{1\over3} = 640320 \qquad  to 30 digits \end{displaymath}
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\samepage \begin{displaymath}640319.99999999999999999999999939032 \end{displaymath}
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\samepage \begin{displaymath}\pi =
\sum_{i=0}^\infty \frac{1}{16^i}\bigr( \frac{4}{8i+1} - \frac{2}{8i+4}-
\frac{1}{8i+5}-\frac{1}{8i+6} \bigl).\end{displaymath}
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{\newpage
\clearpage
\samepage \begin{displaymath}\pi = \int_0^{1/\sqrt2} \frac{4\sqrt2-8x^3-4\sqrt2x^4-8x^5}{1-x^8}dx.\end{displaymath}
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{\newpage
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\samepage \begin{displaymath}\pi =\int_{0}^{1}\!{\frac {16\,y-16}{{y}^{4}-2\,{ y}^{3}+4\,y-4}}{dy}\,.\end{displaymath}
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{\newpage
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\samepage 
\begin{eqnarray*}
\int_0^{1/\sqrt2}\frac{x^{k-1}}{1-x^8}\, dx &=&
 \int_0^{1/\sqrt2}\sum_{i=0}^\infty x^{k-1+8i}\,dx \\
&=& \frac{1}{{\sqrt2}^k} \sum_{i=0}^\infty \frac{1}{16^i(8i+k)}\\
\end{eqnarray*}
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{\newpage
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\samepage \begin{displaymath}p(z) := \sum_{n=0}^N c_nz^n \qquad c_i  \pm 1\end{displaymath}
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\samepage \begin{displaymath}C_1<\frac{|p(z)|}{\sqrt{n}} <C_2 .\end{displaymath}
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