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双曲線関数の無限乗積展開により、以下の式が成り立つ。

\sum_{\begin{smallmatrix}n_i>n_j(i>j) \\ 1\leq i,j<m,n_k\in\mathbb{R}\end{smallmatrix}}\frac{1}{\prod_{k=1}^{m+1}n_k^2}=\frac{\pi^{2m}}{(2m+1)!}

\sum_{\begin{smallmatrix}n_i>n_j(i>j) \\ 1\leq i,j\leq m,n_k\in\mathbb{R}\end{smallmatrix}}\frac{1}{\prod_{k=1}^m(n_k-\frac{1}{2})^2}=\frac{\pi^{2m}}{(2m)!}

この方法はほかの無限積にそのまま応用される。

オイラーの方法でもたらされる級数は積の形に直すことが可能であるが、容易に符号を反転させられるのは周期が4,6,8,12,24の場合だけである。

4の時は4n+1,4n+3

6の時は6n+1,6n+5

8の時は8n+1,8n+3,8n+5,8n+7(2種類の級数の和)

12の時は12n+1,12n+5,12n+7,12n+11(2種類の級数の和)

24の時は24n+1,24n+5,24n+7,24n+11,24n+13,24n+17,24n+19,24n+23(4種類の級数の和)

が分母となる。

また、オイラーの方法でもたらされる関数をarctanのように多項式に治すことはできても意味のある特殊値をもたらすことは実質不可能。

\begin{align}
\prod_{n=1}^\infty\left(1+\frac{k^3}{n^3}\right)=\begin{cases}\frac{\prod_{t=1}^{\frac{k-1}{2}}(k^2-tk+t^2)\cosh\frac{\sqrt{3}}{2}k\pi}{k!\pi}&k\ is\ odd \\
\frac{\frac{\sqrt{3}}{2}k\prod_{t=1}^{\frac{k}{2}-1}(k^2-tk+t^2)\sinh\frac{\sqrt{3}}{2}k\pi}{k!\pi}&k\ is\ even \\
\end{cases} \\
\end{align}

\begin{align}
\prod_{n=1}^\infty\left(1-\frac{x}{n^{2k}}\right)=\begin{cases}
\frac{\sin(\pi \sqrt[2k]{x})}{\pi^k \sqrt{x}2^{\frac{k-1}{2}}}\prod_{m=1}^{\frac{k-1}{2}}\left(\cosh(2\pi \sqrt[2k]{x}\sin\frac{\pi t}{k})-\cos(2\pi \sqrt[2k]{x}\cos\frac{\pi t}{k})\right) \\(k\ is\ odd) \\
\frac{\sin(\pi \sqrt[2k]{x})\sinh(\pi\sqrt[2k]{x})}{\pi^k \sqrt{x}2^{\frac{k}{2}-1}}& \\
\prod_{m=1}^{\frac{k}{2}-1}\left(\cosh(2\pi \sqrt[2k]{x}\sin\frac{\pi t}{k})-\cos(2\pi \sqrt[2k]{x}\cos\frac{\pi t}{k})\right) \\(k\ is\ even) \\
\end{cases} \\
\end{align}

\begin{align}
\prod_{n=m+1}^\infty\left(1-\frac{m^{2k}}{n^{2k}}\right)=\begin{cases}
\frac{1}{2k(m\pi)^{k-1}\prod_{n=1}^{m-1}\left(\frac{m^{2k}}{n^{2k}}-1\right)2^{\frac{k-1}{2}}}\prod_{t=1}^{\frac{k-1}{2}}&\\
\left(\cosh(2\pi m\sin\frac{\pi t}{k})-\cos(2\pi m\cos\frac{\pi t}{k})\right)&k\ is\ odd \\
\frac{\sinh(\pi m)}{2k(m\pi)^{k-1}\prod_{n=1}^{m-1}\left(\frac{m^{2k}}{n^{2k}}-1\right)2^{\frac{k}{2}-1}}\prod_{t=1}^{\frac{k}{2}-1}&\\
\left(\cosh(2\pi m\sin\frac{\pi t}{k})-\cos(2\pi m\cos\frac{\pi t}{k})\right)&k\ is\ even \\
\end{cases} \\
\end{align}

\begin{align}
\prod_{n=2}^\infty\left(1-\frac{1}{n^2}\right)&=\frac{1}{2} \\
\prod_{n=2}^\infty\left(1-\frac{1}{n^3}\right)&=\frac{\cosh\left(\frac{\sqrt{3}}{2}\pi\right)}{3\pi} \\
\prod_{n=2}^\infty\left(1-\frac{1}{n^4}\right)&=\frac{\sinh\pi}{4\pi} \\
\prod_{n=2}^\infty\left(1-\frac{1}{n^6}\right)&=\frac{1+\cosh(\sqrt{3}\pi)}{12\pi^2} \\
\prod_{n=1}^\infty\left(1+\frac{1}{n^2}\right)&=\frac{\sinh\pi}{\pi} \\
\prod_{n=1}^\infty\left(1+\frac{1}{n^3}\right)&=\frac{\cosh\left(\frac{\sqrt{3}}{2}\pi\right)}{\pi} \\
\prod_{n=1}^\infty\left(1+\frac{1}{n^4}\right)&=\frac{\cosh(\pi\sqrt{2})-\cos(\pi\sqrt{2})}{2\pi^2} \\
\prod_{n=1}^\infty\left(1+\frac{1}{n^6}\right)&=\frac{\sinh\pi(\cosh\pi-\cos(\sqrt{3}\pi))}{2\pi^3} \\
\prod_{n=3}^\infty\left(1-\frac{4}{n^2}\right)&=\frac{1}{6} \\
\prod_{n=3}^\infty\left(1-\frac{8}{n^3}\right)&=\frac{\sinh(\pi\sqrt{3})}{42\pi\sqrt{3}} \\
\prod_{n=3}^\infty\left(1-\frac{16}{n^4}\right)&=\frac{\sinh(2\pi)}{120\pi} \\
\end{align}

\begin{align}
\tan \frac{\pi}{4} &=1 \\
\tan \frac{\pi}{6} &=\frac{1}{\sqrt{3}} \\
\tan \frac{\pi}{3} &=\sqrt{3} \\
\tan \frac{\pi}{8} &=\sqrt{2}-1 \\
\tan \frac{3}{8}\pi &=\sqrt{2}+1 \\
\tan \frac{\pi}{10} &=\sqrt{1-\frac{2}{\sqrt{5}}} \\
\tan \frac{\pi}{5} &=\sqrt{5-2\sqrt{5}} \\
\tan \frac{3}{10}\pi &=\sqrt{1+\frac{2}{\sqrt{5}}} \\
\tan \frac{2}{5}\pi &=\sqrt{5+2\sqrt{5}} \\
\tan \frac{\pi}{12} &=2-\sqrt{3} \\
\tan \frac{5}{12}\pi &=2+\sqrt{3} \\
\tan \frac{\pi}{16} &=\sqrt{4+2\sqrt{2}}-\sqrt{2}-1 \\
\tan \frac{3}{16}\pi &=\sqrt{4-2\sqrt{2}}-\sqrt{2}+1 \\
\tan \frac{5}{16}\pi &=\sqrt{4-2\sqrt{2}}+\sqrt{2}-1 \\
\tan \frac{7}{16}\pi &=\sqrt{4+2\sqrt{2}}+\sqrt{2}+1 \\
\end{align}

\sum_{k=1}^\infty \left(\frac{1}{((2k-1)n-m)^s}+\frac{(-1)^s}{((2k-1)n+m)^s}\right)=\frac{P_s \pi^s}{(2n)^s\cdot s!}

\begin{align}
P_1&=k \\
P_2&=2k^2+2 \\
P_3&=6k^3+6k \\
P_4&=24k^4+32k^2+8 \\
P_5&=120k^5+200k^3+80k \\
P_6&=720k^6+1440k^4+816k^2+96 \\
P_7&=5040k^7+11760k^5+8624k^3+1904k \\
\end{align}

\begin{align}
\prod_{n=1}^\infty\left(1+\frac{x}{P(n)}\right)&=\sum_{k=0}^\infty a_kx^k \\
\sum_{n=1}^\infty \frac{1}{P(n)^k}&=b^k \\
\sum_{k=1}^n (-1)^ka_{n-k}b_k&=na_n
\end{align}

\sum_{n=1}^\infty\frac{1}{n^2-a^2}=\frac{\pi}{2a}\cot a\pi -\frac{1}{2a^2}

\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^2-a^2}=\frac{\pi}{2a}\csc a\pi -\frac{1}{2a^2}

\frac{1}{\binom{mn}{pn}}=\int_{0}^1 x^{pn}(1-x)^{(m-p)n}dx

\int x^a(1-x)^b dx=\int \sin^{2a}x \cos^{2b} x 2\sin x\cos xdx

\begin{align}
1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}\cdots&=\frac{1}{4}\pi\\
1-\frac{1}{2}+\frac{1}{4}-\frac{1}{5}+\frac{1}{7}-\frac{1}{8}+\frac{1}{10}-\frac{1}{11}+\frac{1}{13}-\frac{1}{14}\cdots&=\frac{\sqrt{3}}{9}\pi\\
1-\frac{1}{5}+\frac{1}{7}-\frac{1}{11}+\frac{1}{13}-\frac{1}{17}+\frac{1}{19}-\frac{1}{23}+\frac{1}{25}-\frac{1}{29}\cdots&=\frac{\sqrt{3}}{6}\pi\\
\frac{1}{3}-\frac{1}{5}+\frac{1}{11}-\frac{1}{13}+\frac{1}{19}-\frac{1}{21}+\frac{1}{27}-\frac{1}{29}+\frac{1}{35}-\frac{1}{37}\cdots&=\frac{\sqrt{2}-1}{8}\pi\\
1-\frac{1}{7}+\frac{1}{9}-\frac{1}{15}+\frac{1}{17}-\frac{1}{23}+\frac{1}{25}-\frac{1}{31}+\frac{1}{33}-\frac{1}{39}\cdots&=\frac{\sqrt{2}+1}{8}\pi\\
\frac{1}{2}-\frac{1}{3}+\frac{1}{7}-\frac{1}{8}+\frac{1}{12}-\frac{1}{13}+\frac{1}{17}-\frac{1}{18}+\frac{1}{22}-\frac{1}{23}\cdots&=\frac{\sqrt{5-2\sqrt{5}}}{5\sqrt{5}}\pi\\
\frac{1}{3}-\frac{1}{7}+\frac{1}{13}-\frac{1}{17}+\frac{1}{23}-\frac{1}{27}+\frac{1}{33}-\frac{1}{37}+\frac{1}{43}-\frac{1}{47}\cdots&=\frac{\sqrt{5-2\sqrt{5}}}{10}\pi\\
1-\frac{1}{4}+\frac{1}{6}-\frac{1}{9}+\frac{1}{11}-\frac{1}{14}+\frac{1}{16}-\frac{1}{19}+\frac{1}{21}-\frac{1}{24}\cdots&=\frac{\sqrt{5+2\sqrt{5}}}{5\sqrt{5}}\pi\\
1-\frac{1}{9}+\frac{1}{11}-\frac{1}{19}+\frac{1}{21}-\frac{1}{29}+\frac{1}{31}-\frac{1}{39}+\frac{1}{41}-\frac{1}{49}\cdots&=\frac{\sqrt{5+2\sqrt{5}}}{10}\pi\\
\frac{1}{5}-\frac{1}{7}+\frac{1}{17}-\frac{1}{19}+\frac{1}{29}-\frac{1}{31}+\frac{1}{41}-\frac{1}{43}+\frac{1}{53}-\frac{1}{55}\cdots&=\frac{2-\sqrt{3}}{12}\pi\\
1-\frac{1}{11}+\frac{1}{13}-\frac{1}{23}+\frac{1}{25}-\frac{1}{35}+\frac{1}{37}-\frac{1}{47}+\frac{1}{49}-\frac{1}{59}\cdots&=\frac{2+\sqrt{3}}{12}\pi\\
\frac{1}{7}-\frac{1}{9}+\frac{1}{23}-\frac{1}{25}+\frac{1}{39}-\frac{1}{41}+\frac{1}{55}-\frac{1}{57}+\frac{1}{71}-\frac{1}{73}\cdots&=\frac{\sqrt{4+2\sqrt{2}}-\sqrt{2}-1}{16}\pi\\
\frac{1}{5}-\frac{1}{11}+\frac{1}{21}-\frac{1}{27}+\frac{1}{37}-\frac{1}{43}+\frac{1}{53}-\frac{1}{59}+\frac{1}{69}-\frac{1}{75}\cdots&=\frac{\sqrt{4-2\sqrt{2}}-\sqrt{2}+1}{16}\pi\\
\frac{1}{3}-\frac{1}{13}+\frac{1}{19}-\frac{1}{29}+\frac{1}{35}-\frac{1}{45}+\frac{1}{51}-\frac{1}{61}+\frac{1}{67}-\frac{1}{77}\cdots&=\frac{\sqrt{4-2\sqrt{2}}+\sqrt{2}-1}{16}\pi\\
1-\frac{1}{15}+\frac{1}{17}-\frac{1}{31}+\frac{1}{33}-\frac{1}{47}+\frac{1}{49}-\frac{1}{63}+\frac{1}{65}-\frac{1}{79}\cdots&=\frac{\sqrt{4+2\sqrt{2}}+\sqrt{2}+1}{16}\pi\\
\frac{1}{11}-\frac{1}{13}+\frac{1}{35}-\frac{1}{37}+\frac{1}{59}-\frac{1}{61}+\frac{1}{83}-\frac{1}{85}+\frac{1}{107}-\frac{1}{109}\cdots&=\frac{\sqrt{6}-\sqrt{3}+\sqrt{2}-2}{24}\pi\\
\frac{1}{7}-\frac{1}{17}+\frac{1}{31}-\frac{1}{41}+\frac{1}{55}-\frac{1}{65}+\frac{1}{79}-\frac{1}{89}+\frac{1}{103}-\frac{1}{113}\cdots&=\frac{\sqrt{6}+\sqrt{3}-\sqrt{2}-2}{24}\pi\\
\frac{1}{5}-\frac{1}{19}+\frac{1}{29}-\frac{1}{43}+\frac{1}{53}-\frac{1}{67}+\frac{1}{77}-\frac{1}{91}+\frac{1}{101}-\frac{1}{115}\cdots&=\frac{\sqrt{6}+\sqrt{3}+\sqrt{2}+2}{24}\pi\\
1-\frac{1}{23}+\frac{1}{25}-\frac{1}{47}+\frac{1}{49}-\frac{1}{71}+\frac{1}{73}-\frac{1}{95}+\frac{1}{97}-\frac{1}{119}\cdots&=\frac{\sqrt{6}+\sqrt{3}+\sqrt{2}+2}{24}\pi\\
\end{align}

\begin{align}
1+\frac{1}{7}-\frac{1}{9}-\frac{1}{15}+\frac{1}{17}+\frac{1}{23}-\frac{1}{25}-\frac{1}{31}+\frac{1}{33}+\frac{1}{39}\cdots&=\frac{\sqrt{4+2\sqrt{2}}}{8}\pi\\
1-\frac{1}{7}+\frac{1}{9}-\frac{1}{15}+\frac{1}{17}-\frac{1}{23}+\frac{1}{25}-\frac{1}{31}+\frac{1}{33}-\frac{1}{39}\cdots&=\frac{\sqrt{2}+1}{8}\pi\\
\frac{1}{3}+\frac{1}{5}-\frac{1}{11}-\frac{1}{13}+\frac{1}{19}+\frac{1}{21}-\frac{1}{27}-\frac{1}{29}+\frac{1}{35}+\frac{1}{37}\cdots&=\frac{\sqrt{4-2\sqrt{2}}}{8}\pi\\
\frac{1}{3}-\frac{1}{5}+\frac{1}{11}-\frac{1}{13}+\frac{1}{19}-\frac{1}{21}+\frac{1}{27}-\frac{1}{29}+\frac{1}{35}-\frac{1}{37}\cdots&=\frac{\sqrt{2}-1}{8}\pi\\
\frac{1}{7}+\frac{1}{11}-\frac{1}{13}-\frac{1}{17}+\frac{1}{31}+\frac{1}{35}-\frac{1}{37}-\frac{1}{41}+\frac{1}{55}+\frac{1}{59}\cdots&=\frac{\sqrt{6}-2}{12}\pi\\
\frac{1}{7}-\frac{1}{11}+\frac{1}{13}-\frac{1}{17}+\frac{1}{31}-\frac{1}{35}+\frac{1}{37}-\frac{1}{41}+\frac{1}{55}-\frac{1}{59}\cdots&=\frac{\sqrt{3}-\sqrt{2}}{12}\pi\\
1+\frac{1}{5}-\frac{1}{19}-\frac{1}{23}+\frac{1}{25}+\frac{1}{29}-\frac{1}{43}-\frac{1}{47}+\frac{1}{49}+\frac{1}{53}\cdots&=\frac{\sqrt{6}+2}{12}\pi\\
1+\frac{1}{5}-\frac{1}{19}-\frac{1}{23}+\frac{1}{25}+\frac{1}{29}-\frac{1}{43}-\frac{1}{47}+\frac{1}{49}+\frac{1}{53}\cdots&=\frac{\sqrt{3}+\sqrt{2}}{12}\pi\\
\end{align}

\begin{align}
1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\frac{1}{13}-\frac{1}{15}+\frac{1}{17}+\frac{1}{19}\cdots&=\frac{\sqrt{2}}{4}\pi\\
1+\frac{1}{5}-\frac{1}{7}-\frac{1}{11}+\frac{1}{13}+\frac{1}{17}-\frac{1}{19}-\frac{1}{23}+\frac{1}{25}+\frac{1}{29}\cdots&=\frac{\pi}{3}\\
1+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+\frac{1}{6}+\frac{1}{7}-\frac{1}{8}-\frac{1}{9}+\frac{1}{11}+\frac{1}{12}\cdots&=\frac{\sqrt{10+2\sqrt{5}}}{5\sqrt{5}}\pi\\
1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{6}-\frac{1}{7}+\frac{1}{8}-\frac{1}{9}+\frac{1}{11}-\frac{1}{12}\cdots&=\frac{\sqrt{10-2\sqrt{5}}}{5\sqrt{5}}\pi\\
1+\frac{1}{3}-\frac{1}{7}-\frac{1}{9}+\frac{1}{11}+\frac{1}{13}-\frac{1}{17}-\frac{1}{19}+\frac{1}{21}+\frac{1}{23}\cdots&=\frac{\sqrt{10+2\sqrt{5}}}{10}\pi\\
1-\frac{1}{3}+\frac{1}{7}-\frac{1}{9}+\frac{1}{11}-\frac{1}{13}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23}\cdots&=\frac{\sqrt{10-2\sqrt{5}}}{10}\pi\\
1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}-\frac{1}{9}-\frac{1}{11}-\frac{1}{13}-\frac{1}{15}+\frac{1}{17}+\frac{1}{19}\cdots&=\frac{\sqrt{2-\sqrt{2}}}{4}\pi\\
1+\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}-\frac{1}{13}-\frac{1}{15}+\frac{1}{17}+\frac{1}{19}\cdots&=\frac{\sqrt{4-2\sqrt{2}}+\sqrt{2}+1}{8}\pi\\
1+\frac{1}{3}-\frac{1}{5}+\frac{1}{7}-\frac{1}{9}+\frac{1}{11}-\frac{1}{13}-\frac{1}{15}+\frac{1}{17}+\frac{1}{19}\cdots&=\frac{\sqrt{4+2\sqrt{2}}+\sqrt{2}-1}{8}\pi\\
1-\frac{1}{3}+\frac{1}{5}+\frac{1}{7}-\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}\cdots&=\frac{\sqrt{4+2\sqrt{2}}-\sqrt{2}+1}{8}\pi\\
1-\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}\cdots&=\frac{\sqrt{2}+1-\sqrt{4-2\sqrt{2}}}{8}\pi\\
1-\frac{1}{3}-\frac{1}{5}+\frac{1}{7}-\frac{1}{9}+\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}\cdots&=\frac{\sqrt{2-\sqrt{2}}}{4}\pi\\
1+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}-\frac{1}{13}-\frac{1}{17}-\frac{1}{19}-\frac{1}{23}+\frac{1}{25}+\frac{1}{29}\cdots&=\frac{\sqrt{6}}{6}\pi\\
1+\frac{1}{5}+\frac{1}{7}-\frac{1}{11}+\frac{1}{13}-\frac{1}{17}-\frac{1}{19}-\frac{1}{23}+\frac{1}{25}+\frac{1}{29}\cdots&=\frac{\sqrt{6}+\sqrt{3}-\sqrt{2}+2}{12}\pi\\
1+\frac{1}{5}-\frac{1}{7}+\frac{1}{11}-\frac{1}{13}+\frac{1}{17}-\frac{1}{19}-\frac{1}{23}+\frac{1}{25}+\frac{1}{29}\cdots&=\frac{\sqrt{6}-\sqrt{3}+\sqrt{2}+2}{12}\pi\\
1-\frac{1}{5}-\frac{1}{7}-\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}-\frac{1}{23}+\frac{1}{25}-\frac{1}{29}\cdots&=\frac{\sqrt{3}+\sqrt{2}+2-\sqrt{6}}{12}\pi\\
1-\frac{1}{5}-\frac{1}{7}+\frac{1}{11}-\frac{1}{13}+\frac{1}{17}+\frac{1}{19}-\frac{1}{23}+\frac{1}{25}-\frac{1}{29}\cdots&=\frac{\sqrt{2}}{6}\pi\\
1-\frac{1}{5}+\frac{1}{7}+\frac{1}{11}-\frac{1}{13}-\frac{1}{17}+\frac{1}{19}-\frac{1}{23}+\frac{1}{25}-\frac{1}{29}\cdots&=\frac{\sqrt{6}+\sqrt{3}+\sqrt{2}-2}{12}\pi\\
\end{align}


\left(
\begin{matrix}
\tan \frac{\pi}{4}&\tan^3 \frac{\pi}{4}&\tan^5 \frac{\pi}{4}&\tan^7 \frac{\pi}{4}\\
\tan \frac{\pi}{6}&\tan^3 \frac{\pi}{6}&\tan^5 \frac{\pi}{6}&\tan^7 \frac{\pi}{6}\\
\tan \frac{\pi}{3}&\tan^3 \frac{\pi}{3}&\tan^5 \frac{\pi}{3}&\tan^7 \frac{\pi}{3}\\
\tan \frac{\pi}{8}&\tan^3 \frac{\pi}{8}&\tan^5 \frac{\pi}{8}&\tan^7 \frac{\pi}{8}\\
\tan \frac{3\pi}{8}&\tan^3 \frac{3\pi}{8}&\tan^5 \frac{3\pi}{8}&\tan^7 \frac{3\pi}{8}\\
\tan \frac{\pi}{10}&\tan^3 \frac{\pi}{10}&\tan^5 \frac{\pi}{10}&\tan^7 \frac{\pi}{10}\\
\tan \frac{\pi}{5}&\tan^3 \frac{\pi}{5}&\tan^5 \frac{\pi}{5}&\tan^7 \frac{\pi}{5}\\
\tan \frac{3\pi}{10}&\tan^3 \frac{3\pi}{10}&\tan^5 \frac{3\pi}{10}&\tan^7 \frac{3\pi}{10}\\
\tan \frac{2\pi}{5}&\tan^3 \frac{2\pi}{5}&\tan^5 \frac{2\pi}{5}&\tan^7 \frac{2\pi}{5}\\
\tan \frac{\pi}{12}&\tan^3 \frac{\pi}{12}&\tan^5 \frac{\pi}{12}&\tan^7 \frac{\pi}{12}\\
\tan \frac{5\pi}{12}&\tan^3 \frac{5\pi}{12}&\tan^5 \frac{5\pi}{12}&\tan^7 \frac{5\pi}{12}\\
\tan \frac{\pi}{16}&\tan^3 \frac{\pi}{16}&\tan^5 \frac{\pi}{16}&\tan^7 \frac{\pi}{16}\\
\tan \frac{3\pi}{16}&\tan^3 \frac{3\pi}{16}&\tan^5 \frac{3\pi}{16}&\tan^7 \frac{3\pi}{16}\\
\tan \frac{5\pi}{16}&\tan^3 \frac{5\pi}{16}&\tan^5 \frac{5\pi}{16}&\tan^7 \frac{5\pi}{16}\\
\tan \frac{7\pi}{16}&\tan^3 \frac{7\pi}{16}&\tan^5 \frac{7\pi}{16}&\tan^7 \frac{7\pi}{16}\\
\tan \frac{\pi}{24}&\tan^3 \frac{\pi}{24}&\tan^5 \frac{\pi}{24}&\tan^7 \frac{\pi}{24}\\
\tan \frac{5\pi}{24}&\tan^3 \frac{5\pi}{24}&\tan^5 \frac{5\pi}{24}&\tan^7 \frac{5\pi}{24}\\
\tan \frac{7\pi}{24}&\tan^3 \frac{7\pi}{24}&\tan^5 \frac{7\pi}{24}&\tan^7 \frac{7\pi}{24}\\
\tan \frac{11\pi}{24}&\tan^3 \frac{11\pi}{24}&\tan^5 \frac{11\pi}{24}&\tan^7 \frac{11\pi}{24}\\
\end{matrix}
\right) 
\left(
\begin{smallmatrix}
1&1&1&1\\
\frac{1}{\sqrt{3}}&\frac{1}{3\sqrt{3}}&\frac{1}{9\sqrt{3}}&\frac{1}{27\sqrt{3}}\\
\sqrt{3}&3\sqrt{3}&9\sqrt{3}&27\sqrt{3}\\
\sqrt{2}-1&5\sqrt{2}-7&29\sqrt{2}-41&169\sqrt{2}-239\\
\sqrt{2}+1&5\sqrt{2}+7&29\sqrt{2}+41&169\sqrt{2}+239\\
\frac{\sqrt{5}}{5}\sqrt{5-2\sqrt{5}}&\frac{1}{5}(\sqrt{5}-2)\sqrt{5-2\sqrt{5}}&\frac{1}{25}(9\sqrt{5}-20)\sqrt{5-2\sqrt{5}}&\frac{1}{25}(17\sqrt{5}-38)\sqrt{5-2\sqrt{5}}\\
\sqrt{5-2\sqrt{5}}&(5-2\sqrt{5})\sqrt{5-2\sqrt{5}}&5(9-4\sqrt{5})\sqrt{5-2\sqrt{5}}&5(85-38\sqrt{5})\sqrt{5-2\sqrt{5}}\\
\frac{\sqrt{5}}{5}\sqrt{5+2\sqrt{5}}&\frac{1}{5}(\sqrt{5}+2)\sqrt{5+2\sqrt{5}}&\frac{1}{25}(9\sqrt{5}+20)\sqrt{5+2\sqrt{5}}&\frac{1}{25}(17\sqrt{5}+38)\sqrt{5+2\sqrt{5}}\\
\sqrt{5+2\sqrt{5}}&(5+2\sqrt{5})\sqrt{5+2\sqrt{5}}&5(9+4\sqrt{5})\sqrt{5+2\sqrt{5}}&5(85+38\sqrt{5})\sqrt{5+2\sqrt{5}}\\
2-\sqrt{3}&26-15\sqrt{3}&362-209\sqrt{3}&5042-2911\sqrt{3}\\
2+\sqrt{3}&26+15\sqrt{3}&362+209\sqrt{3}&5042+2911\sqrt{3}\\
-1-\sqrt{2}&-31-23\sqrt{2}&-801-569\sqrt{2}&-20287-14351\sqrt{2}\\
+\sqrt{4+2\sqrt{2}}&+\sqrt{2020+1426\sqrt{2}}&+3\sqrt{143236+101282\sqrt{2}}&+\sqrt{823464772+582277474\sqrt{2}}\\
1-\sqrt{2}&31-23\sqrt{2}&801-569\sqrt{2}&20287-14351\sqrt{2}\\
+\sqrt{4-2\sqrt{2}}&+\sqrt{2020-1426\sqrt{2}}&+3\sqrt{143236-101282\sqrt{2}}&+\sqrt{823464772-582277474\sqrt{2}}\\
-1+\sqrt{2}&-31+23\sqrt{2}&-801+569\sqrt{2}&-20287+14351\sqrt{2}\\
+\sqrt{4-2\sqrt{2}}&+\sqrt{2020-1426\sqrt{2}}&+3\sqrt{143236-101282\sqrt{2}}&+\sqrt{823464772-582277474\sqrt{2}}\\
1+\sqrt{2}&31+23\sqrt{2}&801+569\sqrt{2}&20287+14351\sqrt{2}\\
+\sqrt{4+2\sqrt{2}}&+\sqrt{2020+1426\sqrt{2}}&+3\sqrt{143236+101282\sqrt{2}}&+\sqrt{823464772+582277474\sqrt{2}}\\
-2+\sqrt{6}&-110+45\sqrt{6}&-6322+2581\sqrt{6}&-364702+14889\sqrt{6}\\
-\sqrt{3}+\sqrt{2}&-7(9\sqrt{3}-11\sqrt{2})&-41(89\sqrt{3}-109\sqrt{2})&-239(881\sqrt{3}-1079\sqrt{2})\\
-2+\sqrt{6}&-110+45\sqrt{6}&-6322+2581\sqrt{6}&-364702+14889\sqrt{6}\\
+\sqrt{3}-\sqrt{2}&+7(9\sqrt{3}-11\sqrt{2})&+41(89\sqrt{3}-109\sqrt{2})&+239(881\sqrt{3}-1079\sqrt{2})\\
2+\sqrt{6}&110+45\sqrt{6}&6322+2581\sqrt{6}&364702+14889\sqrt{6}\\
-\sqrt{3}-\sqrt{2}&-7(9\sqrt{3}+11\sqrt{2})&-41(89\sqrt{3}+109\sqrt{2})&-239(881\sqrt{3}+1079\sqrt{2})\\
2+\sqrt{6}&110+45\sqrt{6}&6322+2581\sqrt{6}&364702+14889\sqrt{6}\\
+\sqrt{3}+\sqrt{2}&+7(9\sqrt{3}+11\sqrt{2})&+41(89\sqrt{3}+109\sqrt{2})&+239(881\sqrt{3}+1079\sqrt{2})\\
\end{smallmatrix}
\right)


\begin{align}
\sqrt{4+2\sqrt{2}}+\sqrt{4-2\sqrt{2}}&=2\sqrt{2+\sqrt{2}}\\
\sqrt{4+2\sqrt{2}}-\sqrt{4-2\sqrt{2}}&=2\sqrt{2-\sqrt{2}}\\
\sqrt{2020+1426\sqrt{2}}+\sqrt{2020-1426\sqrt{2}}&=2\sqrt{1010+41\sqrt{2}}\\
\sqrt{2020+1426\sqrt{2}}-\sqrt{2020-1426\sqrt{2}}&=2\sqrt{1010-41\sqrt{2}}\\
\sqrt{143236+101282\sqrt{2}}+\sqrt{143236-101282\sqrt{2}}&=2\sqrt{71618+241\sqrt{2}}\\
\sqrt{143236+101282\sqrt{2}}-\sqrt{143236-101282\sqrt{2}}&=2\sqrt{71618-241\sqrt{2}}\\
\sqrt{823464772+582277474\sqrt{2}}&\\
+\sqrt{823464772-582277474\sqrt{2}}&=2\sqrt{411732386+121073\sqrt{2}}&\\
\sqrt{823464772+582277474\sqrt{2}}&\\
-\sqrt{823464772-582277474\sqrt{2}}&=2\sqrt{411732386-121073\sqrt{2}}&\\
\end{align}

\begin{align}
&1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+\frac{1}{9^3}-\frac{1}{11^3}+\frac{1}{13^3}-\frac{1}{15^3}+\frac{1}{17^3}-\frac{1}{19^3}\cdots\\&=\frac{\pi^3}{32}\\
&1-\frac{1}{2^3}+\frac{1}{4^3}-\frac{1}{5^3}+\frac{1}{7^3}-\frac{1}{8^3}+\frac{1}{10^3}-\frac{1}{11^3}+\frac{1}{13^3}-\frac{1}{14^3}\cdots\\&=\frac{\sqrt{3}}{243}\pi^3\\
&1-\frac{1}{5^3}+\frac{1}{7^3}-\frac{1}{11^3}+\frac{1}{13^3}-\frac{1}{17^3}+\frac{1}{19^3}-\frac{1}{23^3}+\frac{1}{25^3}-\frac{1}{29^3}\cdots\\&=\frac{\sqrt{3}}{54}\pi^3\\
&\frac{1}{3^3}-\frac{1}{5^3}+\frac{1}{11^3}-\frac{1}{13^3}+\frac{1}{19^3}-\frac{1}{21^3}+\frac{1}{27^3}-\frac{1}{29^3}+\frac{1}{35^3}-\frac{1}{37^3}\cdots\\&=\frac{3\sqrt{2}-4}{256}\pi^3\\
&1-\frac{1}{7^3}+\frac{1}{9^3}-\frac{1}{15^3}+\frac{1}{17^3}-\frac{1}{23^3}+\frac{1}{25^3}-\frac{1}{31^3}+\frac{1}{33^3}-\frac{1}{39^3}\cdots\\&=\frac{3\sqrt{2}+4}{256}\pi^3\\
&\frac{1}{2^3}-\frac{1}{3^3}+\frac{1}{7^3}-\frac{1}{8^3}+\frac{1}{12^3}-\frac{1}{13^3}+\frac{1}{17^3}-\frac{1}{18^3}+\frac{1}{22^3}-\frac{1}{23^3}\cdots\\&=\frac{(\sqrt{5}-1)\sqrt{5-2\sqrt{5}}}{2500}\pi^3\\
&\frac{1}{3^3}-\frac{1}{7^3}+\frac{1}{13^3}-\frac{1}{17^3}+\frac{1}{23^3}-\frac{1}{27^3}+\frac{1}{33^3}-\frac{1}{37^3}+\frac{1}{43^3}-\frac{1}{47^3}\cdots\\&=\frac{(3-2\sqrt{5})\sqrt{5-2\sqrt{5}}}{500}\pi^3\\
&1-\frac{1}{4^3}+\frac{1}{6^3}-\frac{1}{9^3}+\frac{1}{11^3}-\frac{1}{14^3}+\frac{1}{16^3}-\frac{1}{19^3}+\frac{1}{21^3}-\frac{1}{24^3}\cdots\\&=\frac{(\sqrt{5}+1)\sqrt{5+2\sqrt{5}}}{2500}\pi^3\\
&1-\frac{1}{9^3}+\frac{1}{11^3}-\frac{1}{19^3}+\frac{1}{21^3}-\frac{1}{29^3}+\frac{1}{31^3}-\frac{1}{39^3}+\frac{1}{41^3}-\frac{1}{49^3}\cdots\\&=\frac{(3+2\sqrt{5})\sqrt{5+2\sqrt{5}}}{500}\pi^3\\
&\frac{1}{5^3}-\frac{1}{7^3}+\frac{1}{17^3}-\frac{1}{19^3}+\frac{1}{29^3}-\frac{1}{31^3}+\frac{1}{41^3}-\frac{1}{43^3}+\frac{1}{53^3}-\frac{1}{55^3}\cdots\\&=\frac{7-4\sqrt{3}}{432}\pi^3\\
&1-\frac{1}{11^3}+\frac{1}{13^3}-\frac{1}{23^3}+\frac{1}{25^3}-\frac{1}{35^3}+\frac{1}{37^3}-\frac{1}{47^3}+\frac{1}{49^3}-\frac{1}{59^3}\cdots\\&=\frac{7+4\sqrt{3}}{432}\pi^3\\
&\frac{1}{7^3}-\frac{1}{9^3}+\frac{1}{23^3}-\frac{1}{25^3}+\frac{1}{39^3}-\frac{1}{41^3}+\frac{1}{55^3}-\frac{1}{57^3}+\frac{1}{71^3}-\frac{1}{73^3}\cdots\\&=\frac{\sqrt{548+286\sqrt{2}}-16-12\sqrt{2}}{2048}\pi^3\\
&\frac{1}{5^3}-\frac{1}{11^3}+\frac{1}{21^3}-\frac{1}{27^3}+\frac{1}{37^3}-\frac{1}{43^3}+\frac{1}{53^3}-\frac{1}{59^3}+\frac{1}{69^3}-\frac{1}{75^3}\cdots\\&=\frac{\sqrt{548-286\sqrt{2}}+16-12\sqrt{2}}{2048}\pi^3\\
&\frac{1}{3^3}-\frac{1}{13^3}+\frac{1}{19^3}-\frac{1}{29^3}+\frac{1}{35^3}-\frac{1}{45^3}+\frac{1}{51^3}-\frac{1}{61^3}+\frac{1}{67^3}-\frac{1}{77^3}\cdots\\&=\frac{\sqrt{548-286\sqrt{2}}-16+12\sqrt{2}}{2048}\pi^3\\
&1-\frac{1}{15^3}+\frac{1}{17^3}-\frac{1}{31^3}+\frac{1}{33^3}-\frac{1}{47^3}+\frac{1}{49^3}-\frac{1}{63^3}+\frac{1}{65^3}-\frac{1}{79^3}\cdots\\&=\frac{\sqrt{548+286\sqrt{2}}+16+12\sqrt{2}}{2048}\pi^3\\
&\frac{1}{11^3}-\frac{1}{13^3}+\frac{1}{35^3}-\frac{1}{37^3}+\frac{1}{59^3}-\frac{1}{61^3}+\frac{1}{83^3}-\frac{1}{85^3}+\frac{1}{107^3}-\frac{1}{109^3}\cdots\\&=\frac{23\sqrt{6}-32\sqrt{3}+39\sqrt{2}-56}{6912}\pi^3\\
&\frac{1}{7^3}-\frac{1}{17^3}+\frac{1}{31^3}-\frac{1}{41^3}+\frac{1}{55^3}-\frac{1}{65^3}+\frac{1}{79^3}-\frac{1}{89^3}+\frac{1}{103^3}-\frac{1}{113^3}\cdots\\&=\frac{23\sqrt{6}+32\sqrt{3}-39\sqrt{2}-56}{6912}\pi^3\\
&\frac{1}{5^3}-\frac{1}{19^3}+\frac{1}{29^3}-\frac{1}{43^3}+\frac{1}{53^3}-\frac{1}{67^3}+\frac{1}{77^3}-\frac{1}{91^3}+\frac{1}{101^3}-\frac{1}{115^3}\cdots\\&=\frac{23\sqrt{6}-32\sqrt{3}-39\sqrt{2}+56}{6912}\pi^3\\
&1-\frac{1}{23^3}+\frac{1}{25^3}-\frac{1}{47^3}+\frac{1}{49^3}-\frac{1}{71^3}+\frac{1}{73^3}-\frac{1}{95^3}+\frac{1}{97^3}-\frac{1}{119^3}\cdots\\&=\frac{23\sqrt{6}+32\sqrt{3}+39\sqrt{2}+56}{6912}\pi^3\\
\end{align}


\begin{align}
&1+\frac{1}{3^3}-\frac{1}{5^3}-\frac{1}{7^3}+\frac{1}{9^3}+\frac{1}{11^3}-\frac{1}{13^3}-\frac{1}{15^3}+\frac{1}{17^3}+\frac{1}{19^3}\cdots\\&=\frac{3\sqrt{2}}{128}\pi^3\\
&1+\frac{1}{5^3}-\frac{1}{7^3}-\frac{1}{11^3}+\frac{1}{13^3}+\frac{1}{17^3}-\frac{1}{19^3}-\frac{1}{23^3}+\frac{1}{25^3}+\frac{1}{29^3}\cdots\\&=\frac{7}{216}\pi^3\\
&1+\frac{1}{2^3}-\frac{1}{3^3}-\frac{1}{4^3}+\frac{1}{6^3}+\frac{1}{7^3}-\frac{1}{8^3}-\frac{1}{9^3}+\frac{1}{11^3}+\frac{1}{12^3}\cdots\\&=\frac{\sqrt{10(5+\sqrt{5})}+\sqrt{2(5-\sqrt{5})}}{2500}\pi^3\\
&1-\frac{1}{2^3}+\frac{1}{3^3}-\frac{1}{4^3}+\frac{1}{6^3}-\frac{1}{7^3}+\frac{1}{8^3}-\frac{1}{9^3}+\frac{1}{11^3}-\frac{1}{12^3}\cdots\\&=\frac{\sqrt{10(5-\sqrt{5})}+\sqrt{2(5+\sqrt{5})}}{2500}\pi^3\\
&1+\frac{1}{3^3}-\frac{1}{7^3}-\frac{1}{9^3}+\frac{1}{11^3}+\frac{1}{13^3}-\frac{1}{17^3}-\frac{1}{19^3}+\frac{1}{21^3}+\frac{1}{23^3}\cdots\\&=\frac{2\sqrt{10(5+\sqrt{5})}+3\sqrt{2(5-\sqrt{5})}}{500}\pi^3\\
&1-\frac{1}{3^3}+\frac{1}{7^3}-\frac{1}{9^3}+\frac{1}{11^3}-\frac{1}{13^3}+\frac{1}{17^3}-\frac{1}{19^3}+\frac{1}{21^3}-\frac{1}{23^3}\cdots\\&=\frac{2\sqrt{10(5-\sqrt{5})}+3\sqrt{2(5+\sqrt{5})}}{500}\pi^3\\
&1+\frac{1}{3^3}+\frac{1}{5^3}+\frac{1}{7^3}-\frac{1}{9^3}-\frac{1}{11^3}-\frac{1}{13^3}-\frac{1}{15^3}+\frac{1}{17^3}+\frac{1}{19^3}\cdots\\&=\frac{\sqrt{274+\sqrt{34178}}}{1024}\pi\\
&1+\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+\frac{1}{9^3}-\frac{1}{11^3}-\frac{1}{13^3}-\frac{1}{15^3}+\frac{1}{17^3}+\frac{1}{19^3}\cdots\\&=\frac{\sqrt{548-286\sqrt{2}}+16\sqrt{2}+12}{2048}\pi\\
&1+\frac{1}{3^3}-\frac{1}{5^3}+\frac{1}{7^3}-\frac{1}{9^3}+\frac{1}{11^3}-\frac{1}{13^3}-\frac{1}{15^3}+\frac{1}{17^3}+\frac{1}{19^3}\cdots\\&=\frac{\sqrt{548+286\sqrt{2}}+16\sqrt{2}-12}{2048}\pi\\
&1-\frac{1}{3^3}+\frac{1}{5^3}+\frac{1}{7^3}-\frac{1}{9^3}-\frac{1}{11^3}+\frac{1}{13^3}-\frac{1}{15^3}+\frac{1}{17^3}-\frac{1}{19^3}\cdots\\&=\frac{\sqrt{548+286\sqrt{2}}-16\sqrt{2}+12}{2048}\pi\\
&1-\frac{1}{3^3}-\frac{1}{5^3}-\frac{1}{7^3}+\frac{1}{9^3}+\frac{1}{11^3}+\frac{1}{13^3}-\frac{1}{15^3}+\frac{1}{17^3}-\frac{1}{19^3}\cdots\\&=\frac{16\sqrt{2}+12-\sqrt{548-286\sqrt{2}}}{2048}\pi\\
&1-\frac{1}{3^3}-\frac{1}{5^3}+\frac{1}{7^3}-\frac{1}{9^3}+\frac{1}{11^3}+\frac{1}{13^3}-\frac{1}{15^3}+\frac{1}{17^3}-\frac{1}{19^3}\cdots\\&=\frac{\sqrt{274+\sqrt{34178}}}{1024}\pi\\
&1+\frac{1}{5^3}+\frac{1}{7^3}+\frac{1}{11^3}-\frac{1}{13^3}-\frac{1}{17^3}-\frac{1}{19^3}-\frac{1}{23^3}+\frac{1}{25^3}+\frac{1}{29^3}\cdots\\&=\frac{23\sqrt{6}}{1728}\pi^3\\
&1+\frac{1}{5^3}+\frac{1}{7^3}-\frac{1}{11^3}+\frac{1}{13^3}-\frac{1}{17^3}-\frac{1}{19^3}-\frac{1}{23^3}+\frac{1}{25^3}+\frac{1}{29^3}\cdots\\&=\frac{23\sqrt{6}+32\sqrt{3}-39\sqrt{2}+56}{3456}\pi^3\\
&1+\frac{1}{5^3}-\frac{1}{7^3}+\frac{1}{11^3}-\frac{1}{13^3}+\frac{1}{17^3}-\frac{1}{19^3}-\frac{1}{23^3}+\frac{1}{25^3}+\frac{1}{29^3}\cdots\\&=\frac{23\sqrt{6}-32\sqrt{3}+39\sqrt{2}+56}{3456}\pi^3\\
&1-\frac{1}{5^3}-\frac{1}{7^3}-\frac{1}{11^3}+\frac{1}{13^3}+\frac{1}{17^3}+\frac{1}{19^3}-\frac{1}{23^3}+\frac{1}{25^3}-\frac{1}{29^3}\cdots\\&=\frac{-23\sqrt{6}+32\sqrt{3}+39\sqrt{2}+56}{3456}\pi^3\\
&1-\frac{1}{5^3}-\frac{1}{7^3}+\frac{1}{11^3}-\frac{1}{13^3}+\frac{1}{17^3}+\frac{1}{19^3}-\frac{1}{23^3}+\frac{1}{25^3}-\frac{1}{29^3}\cdots\\&=\frac{13\sqrt{2}}{576}\pi\\
&1-\frac{1}{5^3}+\frac{1}{7^3}+\frac{1}{11^3}-\frac{1}{13^3}-\frac{1}{17^3}+\frac{1}{19^3}-\frac{1}{23^3}+\frac{1}{25^3}-\frac{1}{29^3}\cdots\\&=\frac{23\sqrt{6}+32\sqrt{3}+39\sqrt{2}-56}{3456}\pi^3\\
\end{align}


\begin{align}
&1-\frac{1}{3^5}+\frac{1}{5^5}-\frac{1}{7^5}+\frac{1}{9^5}-\frac{1}{11^5}+\frac{1}{13^5}-\frac{1}{15^5}+\frac{1}{17^5}-\frac{1}{19^5}\cdots\\&=\frac{5}{1596}\pi^5\\
&1-\frac{1}{2^5}+\frac{1}{4^5}-\frac{1}{5^5}+\frac{1}{7^5}-\frac{1}{8^5}+\frac{1}{10^5}-\frac{1}{11^5}+\frac{1}{13^5}-\frac{1}{14^5}\cdots\\&=\frac{\sqrt{3}}{17496}\pi^5\\
&1-\frac{1}{5^5}+\frac{1}{7^5}-\frac{1}{11^5}+\frac{1}{13^5}-\frac{1}{17^5}+\frac{1}{19^5}-\frac{1}{23^5}+\frac{1}{25^5}-\frac{1}{29^5}\cdots\\&=\frac{11\sqrt{3}}{5832}\pi^5\\
&\frac{1}{3^5}-\frac{1}{5^5}+\frac{1}{11^5}-\frac{1}{13^5}+\frac{1}{19^5}-\frac{1}{21^5}+\frac{1}{27^5}-\frac{1}{29^5}+\frac{1}{35^5}-\frac{1}{37^5}\cdots\\&=\frac{57\sqrt{2}-80}{49152}\pi^5\\
&1-\frac{1}{7^5}+\frac{1}{9^5}-\frac{1}{15^5}+\frac{1}{17^5}-\frac{1}{23^5}+\frac{1}{25^5}-\frac{1}{31^5}+\frac{1}{33^5}-\frac{1}{39^5}\cdots\\&=\frac{57\sqrt{2}+80}{49152}\pi^5\\
&\frac{1}{2^5}-\frac{1}{3^5}+\frac{1}{7^5}-\frac{1}{8^5}+\frac{1}{12^5}-\frac{1}{13^5}+\frac{1}{17^5}-\frac{1}{18^5}+\frac{1}{22^5}-\frac{1}{23^5}\cdots\\&=\frac{(31\sqrt{5}-55)\sqrt{5-2\sqrt{5}}}{3750000}\pi^5\\
&\frac{1}{3^5}-\frac{1}{7^5}+\frac{1}{13^5}-\frac{1}{17^5}+\frac{1}{23^5}-\frac{1}{27^5}+\frac{1}{33^5}-\frac{1}{37^5}+\frac{1}{43^5}-\frac{1}{47^5}\cdots\\&=\frac{(81-35\sqrt{5})\sqrt{5-2\sqrt{5}}}{150000}\pi^5\\
&1-\frac{1}{4^5}+\frac{1}{6^5}-\frac{1}{9^5}+\frac{1}{11^5}-\frac{1}{14^5}+\frac{1}{16^5}-\frac{1}{19^5}+\frac{1}{21^5}-\frac{1}{24^5}\cdots\\&=\frac{(31\sqrt{5}+55)\sqrt{5+2\sqrt{5}}}{3750000}\pi^5\\
&1-\frac{1}{9^5}+\frac{1}{11^5}-\frac{1}{19^5}+\frac{1}{21^5}-\frac{1}{29^5}+\frac{1}{31^5}-\frac{1}{39^5}+\frac{1}{41^5}-\frac{1}{49^5}\cdots\\&=\frac{(81+35\sqrt{5})\sqrt{5+2\sqrt{5}}}{150000}\pi^5\\
&\frac{1}{5^5}-\frac{1}{7^5}+\frac{1}{17^5}-\frac{1}{19^5}+\frac{1}{29^5}-\frac{1}{31^5}+\frac{1}{41^5}-\frac{1}{43^5}+\frac{1}{53^5}-\frac{1}{55^5}\cdots\\&=\frac{305-176\sqrt{3}}{186624}\pi^5\\
&1-\frac{1}{11^5}+\frac{1}{13^5}-\frac{1}{23^5}+\frac{1}{25^5}-\frac{1}{35^5}+\frac{1}{37^5}-\frac{1}{47^5}+\frac{1}{49^5}-\frac{1}{59^5}\cdots\\&=\frac{305+176\sqrt{3}}{186624}\pi^5\\
&\frac{1}{7^5}-\frac{1}{9^5}+\frac{1}{23^5}-\frac{1}{25^5}+\frac{1}{39^5}-\frac{1}{41^5}+\frac{1}{55^5}-\frac{1}{57^5}+\frac{1}{71^5}-\frac{1}{73^5}\cdots\\&=\frac{\sqrt{3302468+2335106\sqrt{2}}-1280-912\sqrt{2}}{1572864}\pi^5\\
&\frac{1}{5^5}-\frac{1}{11^5}+\frac{1}{21^5}-\frac{1}{27^5}+\frac{1}{37^5}-\frac{1}{43^5}+\frac{1}{53^5}-\frac{1}{59^5}+\frac{1}{69^5}-\frac{1}{75^5}\cdots\\&=\frac{\sqrt{3302468-2335106\sqrt{2}}+1280-912\sqrt{2}}{1572864}\pi^5\\
&\frac{1}{3^5}-\frac{1}{13^5}+\frac{1}{19^5}-\frac{1}{29^5}+\frac{1}{35^5}-\frac{1}{45^5}+\frac{1}{51^5}-\frac{1}{61^5}+\frac{1}{67^5}-\frac{1}{77^5}\cdots\\&=\frac{\sqrt{3302468-2335106\sqrt{2}}-1280+912\sqrt{2}}{1572864}\pi^5\\
&1-\frac{1}{15^5}+\frac{1}{17^5}-\frac{1}{31^5}+\frac{1}{33^5}-\frac{1}{47^5}+\frac{1}{49^5}-\frac{1}{63^5}+\frac{1}{65^5}-\frac{1}{79^5}\cdots\\&=\frac{\sqrt{3302468+2335106\sqrt{2}}+1280+912\sqrt{2}}{1572864}\pi^5\\
&\frac{1}{11^5}-\frac{1}{13^5}+\frac{1}{35^5}-\frac{1}{37^5}+\frac{1}{59^5}-\frac{1}{61^5}+\frac{1}{83^5}-\frac{1}{85^5}+\frac{1}{107^5}-\frac{1}{109^5}\cdots\\&=\frac{3985\sqrt{6}-5632\sqrt{3}+6897\sqrt{2}-9760}{11943936}\pi^5\\
&\frac{1}{7^5}-\frac{1}{17^5}+\frac{1}{31^5}-\frac{1}{41^5}+\frac{1}{55^5}-\frac{1}{65^5}+\frac{1}{79^5}-\frac{1}{89^5}+\frac{1}{103^5}-\frac{1}{113^5}\cdots\\&=\frac{3985\sqrt{6}+5632\sqrt{3}-6897\sqrt{2}-9760}{11943936}\pi^5i\\
&\frac{1}{5^5}-\frac{1}{19^5}+\frac{1}{29^5}-\frac{1}{43^5}+\frac{1}{53^5}-\frac{1}{67^5}+\frac{1}{77^5}-\frac{1}{91^5}+\frac{1}{101^5}-\frac{1}{115^5}\cdots\\&=\frac{3985\sqrt{6}-5632\sqrt{3}-6897\sqrt{2}+9760}{11943936}\pi^5\\
&1-\frac{1}{23^5}+\frac{1}{25^5}-\frac{1}{47^5}+\frac{1}{49^5}-\frac{1}{71^5}+\frac{1}{73^5}-\frac{1}{95^5}+\frac{1}{97^5}-\frac{1}{119^5}\cdots\\&=\frac{3985\sqrt{6}+5632\sqrt{3}+6897\sqrt{2}+9760}{11943936}\pi^5\\
\end{align}


\begin{align}
&1+\frac{1}{3^5}-\frac{1}{5^5}-\frac{1}{7^5}+\frac{1}{9^5}+\frac{1}{11^5}-\frac{1}{13^5}-\frac{1}{15^5}+\frac{1}{17^5}+\frac{1}{19^5}\cdots\\&=\frac{\sqrt{2}}{4}\pi\\
&1+\frac{1}{5^5}-\frac{1}{7^5}-\frac{1}{11^5}+\frac{1}{13^5}+\frac{1}{17^5}-\frac{1}{19^5}-\frac{1}{23^5}+\frac{1}{25^5}+\frac{1}{29^5}\cdots\\&=\frac{\pi}{3}\\
&1+\frac{1}{2^5}-\frac{1}{3^5}-\frac{1}{4^5}+\frac{1}{6^5}+\frac{1}{7^5}-\frac{1}{8^5}-\frac{1}{9^5}+\frac{1}{11^5}+\frac{1}{12^5}\cdots\\&=\frac{\sqrt{10+2\sqrt{5}}}{5\sqrt{5}}\pi\\
&1+\frac{1}{2^5}+\frac{1}{3^5}-\frac{1}{4^5}+\frac{1}{6^5}-\frac{1}{7^5}+\frac{1}{8^5}-\frac{1}{9^5}+\frac{1}{11^5}-\frac{1}{12^5}\cdots\\&=\frac{\sqrt{10-2\sqrt{5}}}{5\sqrt{5}}\pi\\
&1+\frac{1}{3^5}-\frac{1}{7^5}-\frac{1}{9^5}+\frac{1}{11^5}+\frac{1}{13^5}-\frac{1}{17^5}-\frac{1}{19^5}+\frac{1}{21^5}+\frac{1}{23^5}\cdots\\&=\frac{\sqrt{10+2\sqrt{5}}}{10}\pi\\
&1+\frac{1}{3^5}+\frac{1}{7^5}-\frac{1}{9^5}+\frac{1}{11^5}-\frac{1}{13^5}+\frac{1}{17^5}-\frac{1}{19^5}+\frac{1}{21^5}-\frac{1}{23^5}\cdots\\&=\frac{\sqrt{10-2\sqrt{5}}}{10}\pi\\
&1+\frac{1}{3^5}+\frac{1}{5^5}+\frac{1}{7^5}-\frac{1}{9^5}-\frac{1}{11^5}-\frac{1}{13^5}-\frac{1}{15^5}+\frac{1}{17^5}+\frac{1}{19^5}\cdots\\&=\frac{\sqrt{2-\sqrt{2}}}{4}\pi\\
&1+\frac{1}{3^5}+\frac{1}{5^5}-\frac{1}{7^5}+\frac{1}{9^5}-\frac{1}{11^5}-\frac{1}{13^5}-\frac{1}{15^5}+\frac{1}{17^5}+\frac{1}{19^5}\cdots\\&=\frac{\sqrt{4-2\sqrt{2}}+\sqrt{2}+1}{8}\pi\\
&1+\frac{1}{3^5}-\frac{1}{5^5}+\frac{1}{7^5}-\frac{1}{9^5}+\frac{1}{11^5}-\frac{1}{13^5}-\frac{1}{15^5}+\frac{1}{17^5}+\frac{1}{19^5}\cdots\\&=\frac{\sqrt{4+2\sqrt{2}}+\sqrt{2}-1}{8}\pi\\
&1+\frac{1}{3^5}+\frac{1}{5^5}+\frac{1}{7^5}-\frac{1}{9^5}-\frac{1}{11^5}+\frac{1}{13^5}-\frac{1}{15^5}+\frac{1}{17^5}-\frac{1}{19^5}\cdots\\&=\frac{\sqrt{4+2\sqrt{2}}-\sqrt{2}+1}{8}\pi\\
&1+\frac{1}{3^5}-\frac{1}{5^5}-\frac{1}{7^5}+\frac{1}{9^5}+\frac{1}{11^5}+\frac{1}{13^5}-\frac{1}{15^5}+\frac{1}{17^5}-\frac{1}{19^5}\cdots\\&=\frac{\sqrt{2}+1+\sqrt{4-2\sqrt{2}}}{8}\pi\\
&1+\frac{1}{3^5}-\frac{1}{5^5}+\frac{1}{7^5}-\frac{1}{9^5}+\frac{1}{11^5}+\frac{1}{13^5}-\frac{1}{15^5}+\frac{1}{17^5}-\frac{1}{19^5}\cdots\\&=\frac{\sqrt{2-\sqrt{2}}}{4}\pi\\
&1+\frac{1}{5^5}+\frac{1}{7^5}+\frac{1}{11^5}-\frac{1}{13^5}-\frac{1}{17^5}-\frac{1}{19^5}-\frac{1}{23^5}+\frac{1}{25^5}+\frac{1}{29^5}\cdots\\&=\frac{\sqrt{6}}{6}\pi\\
&1+\frac{1}{5^5}+\frac{1}{7^5}-\frac{1}{11^5}+\frac{1}{13^5}-\frac{1}{17^5}-\frac{1}{19^5}-\frac{1}{23^5}+\frac{1}{25^5}+\frac{1}{29^5}\cdots\\&=\frac{\sqrt{6}+\sqrt{3}-\sqrt{2}+2}{12}\pi\\
&1+\frac{1}{5^5}-\frac{1}{7^5}+\frac{1}{11^5}-\frac{1}{13^5}+\frac{1}{17^5}-\frac{1}{19^5}-\frac{1}{23^5}+\frac{1}{25^5}+\frac{1}{29^5}\cdots\\&=\frac{\sqrt{6}-\sqrt{3}+\sqrt{2}+2}{12}\pi\\
&1+\frac{1}{5^5}-\frac{1}{7^5}-\frac{1}{11^5}+\frac{1}{13^5}+\frac{1}{17^5}+\frac{1}{19^5}-\frac{1}{23^5}+\frac{1}{25^5}-\frac{1}{29^5}\cdots\\&=\frac{\sqrt{3}+\sqrt{2}+2-\sqrt{6}}{12}\pi\\
&1+\frac{1}{5^5}-\frac{1}{7^5}+\frac{1}{11^5}-\frac{1}{13^5}+\frac{1}{17^5}+\frac{1}{19^5}-\frac{1}{23^5}+\frac{1}{25^5}-\frac{1}{29^5}\cdots\\&=\frac{\sqrt{2}}{6}\pi\\
&1+\frac{1}{5^5}+\frac{1}{7^5}+\frac{1}{11^5}-\frac{1}{13^5}-\frac{1}{17^5}+\frac{1}{19^5}-\frac{1}{23^5}+\frac{1}{25^5}-\frac{1}{29^5}\cdots\\&=\frac{\sqrt{6}+\sqrt{3}+\sqrt{2}-2}{12}\pi\\
\end{align}


\begin{align}
&1-\frac{1}{3^7}+\frac{1}{5^7}-\frac{1}{7^7}+\frac{1}{9^7}-\frac{1}{11^7}+\frac{1}{13^7}-\frac{1}{15^7}+\frac{1}{17^7}-\frac{1}{19^7}\cdots\\&=\frac{1}{4}\pi\\
&1-\frac{1}{2^7}+\frac{1}{4^7}-\frac{1}{5^7}+\frac{1}{7^7}-\frac{1}{8^7}+\frac{1}{10^7}-\frac{1}{11^7}+\frac{1}{13^7}-\frac{1}{14^7}\cdots\\&=\frac{\sqrt{3}}{9}\pi\\
&1-\frac{1}{5^7}+\frac{1}{7^7}-\frac{1}{11^7}+\frac{1}{13^7}-\frac{1}{17^7}+\frac{1}{19^7}-\frac{1}{23^7}+\frac{1}{25^7}-\frac{1}{29^7}\cdots\\&=\frac{\sqrt{3}}{6}\pi\\
&\frac{1}{3^7}-\frac{1}{5^7}+\frac{1}{11^7}-\frac{1}{13^7}+\frac{1}{19^7}-\frac{1}{21^7}+\frac{1}{27^7}-\frac{1}{29^7}+\frac{1}{35^7}-\frac{1}{37^7}\cdots\\&=\frac{\sqrt{2}-1}{8}\pi\\
&1-\frac{1}{7^7}+\frac{1}{9^7}-\frac{1}{15^7}+\frac{1}{17^7}-\frac{1}{23^7}+\frac{1}{25^7}-\frac{1}{31^7}+\frac{1}{33^7}-\frac{1}{39^7}\cdots\\&=\frac{\sqrt{2}+1}{8}\pi\\
&\frac{1}{2^7}-\frac{1}{3^7}+\frac{1}{7^7}-\frac{1}{8^7}+\frac{1}{12^7}-\frac{1}{13^7}+\frac{1}{17^7}-\frac{1}{18^7}+\frac{1}{22^7}-\frac{1}{23^7}\cdots\\&=\frac{\sqrt{5-2\sqrt{5}}}{5\sqrt{5}}\pi\\
&\frac{1}{3^7}-\frac{1}{7^7}+\frac{1}{13^7}-\frac{1}{17^7}+\frac{1}{23^7}-\frac{1}{27^7}+\frac{1}{33^7}-\frac{1}{37^7}+\frac{1}{43^7}-\frac{1}{47^7}\cdots\\&=\frac{\sqrt{5-2\sqrt{5}}}{10}\pi\\
&1-\frac{1}{4^7}+\frac{1}{6^7}-\frac{1}{9^7}+\frac{1}{11^7}-\frac{1}{14^7}+\frac{1}{16^7}-\frac{1}{19^7}+\frac{1}{21^7}-\frac{1}{24^7}\cdots\\&=\frac{\sqrt{5+2\sqrt{5}}}{5\sqrt{5}}\pi\\
&1-\frac{1}{9^7}+\frac{1}{11^7}-\frac{1}{19^7}+\frac{1}{21^7}-\frac{1}{29^7}+\frac{1}{31^7}-\frac{1}{39^7}+\frac{1}{41^7}-\frac{1}{49^7}\cdots\\&=\frac{\sqrt{5+2\sqrt{5}}}{10}\pi\\
&\frac{1}{5^7}-\frac{1}{7^7}+\frac{1}{17^7}-\frac{1}{19^7}+\frac{1}{29^7}-\frac{1}{31^7}+\frac{1}{41^7}-\frac{1}{43^7}+\frac{1}{53^7}-\frac{1}{55^7}\cdots\\&=\frac{2-\sqrt{3}}{12}\pi\\
&1-\frac{1}{11^7}+\frac{1}{13^7}-\frac{1}{23^7}+\frac{1}{25^7}-\frac{1}{35^7}+\frac{1}{37^7}-\frac{1}{47^7}+\frac{1}{49^7}-\frac{1}{59^7}\cdots\\&=\frac{2+\sqrt{3}}{12}\pi\\
&\frac{1}{7^7}-\frac{1}{9^7}+\frac{1}{23^7}-\frac{1}{25^7}+\frac{1}{39^7}-\frac{1}{41^7}+\frac{1}{55^7}-\frac{1}{57^7}+\frac{1}{71^7}-\frac{1}{73^7}\cdots\\&=\frac{\sqrt{4+2\sqrt{2}}-\sqrt{2}-1}{16}\pi\\
&\frac{1}{5^7}-\frac{1}{11^7}+\frac{1}{21^7}-\frac{1}{27^7}+\frac{1}{37^7}-\frac{1}{43^7}+\frac{1}{53^7}-\frac{1}{59^7}+\frac{1}{69^7}-\frac{1}{75^7}\cdots\\&=\frac{\sqrt{4-2\sqrt{2}}-\sqrt{2}+1}{16}\pi\\
&\frac{1}{3^7}-\frac{1}{13^7}+\frac{1}{19^7}-\frac{1}{29^7}+\frac{1}{35^7}-\frac{1}{45^7}+\frac{1}{51^7}-\frac{1}{61^7}+\frac{1}{67^7}-\frac{1}{77^7}\cdots\\&=\frac{\sqrt{4-2\sqrt{2}}+\sqrt{2}-1}{16}\pi\\
&1-\frac{1}{15^7}+\frac{1}{17^7}-\frac{1}{31^7}+\frac{1}{33^7}-\frac{1}{47^7}+\frac{1}{49^7}-\frac{1}{63^7}+\frac{1}{65^7}-\frac{1}{79^7}\cdots\\&=\frac{\sqrt{4+2\sqrt{2}}+\sqrt{2}+1}{16}\pi\\
&\frac{1}{11^7}-\frac{1}{13^7}+\frac{1}{35^7}-\frac{1}{37^7}+\frac{1}{59^7}-\frac{1}{61^7}+\frac{1}{83^7}-\frac{1}{85^7}+\frac{1}{107^7}-\frac{1}{109^7}\cdots\\&=\frac{\sqrt{6}-\sqrt{3}+\sqrt{2}-2}{24}\pi\\
&\frac{1}{7^7}-\frac{1}{17^7}+\frac{1}{31^7}-\frac{1}{41^7}+\frac{1}{55^7}-\frac{1}{65^7}+\frac{1}{79^7}-\frac{1}{89^7}+\frac{1}{103^7}-\frac{1}{113^7}\cdots\\&=\frac{\sqrt{6}+\sqrt{3}-\sqrt{2}-2}{24}\pi\\
&\frac{1}{5^7}-\frac{1}{19^7}+\frac{1}{29^7}-\frac{1}{43^7}+\frac{1}{53^7}-\frac{1}{67^7}+\frac{1}{77^7}-\frac{1}{91^7}+\frac{1}{101^7}-\frac{1}{115^7}\cdots\\&=\frac{\sqrt{6}+\sqrt{3}+\sqrt{2}+2}{24}\pi\\
&1-\frac{1}{23^7}+\frac{1}{25^7}-\frac{1}{47^7}+\frac{1}{49^7}-\frac{1}{71^7}+\frac{1}{73^7}-\frac{1}{95^7}+\frac{1}{97^7}-\frac{1}{119^7}\cdots\\&=\frac{\sqrt{6}+\sqrt{3}+\sqrt{2}+2}{24}\pi\\
\end{align}


\begin{align}
&1+\frac{1}{3^7}-\frac{1}{5^7}-\frac{1}{7^7}+\frac{1}{9^7}+\frac{1}{11^7}-\frac{1}{13^7}-\frac{1}{15^7}+\frac{1}{17^7}+\frac{1}{19^7}\cdots\\&=\frac{\sqrt{2}}{4}\pi\\
&1+\frac{1}{5^7}-\frac{1}{7^7}-\frac{1}{11^7}+\frac{1}{13^7}+\frac{1}{17^7}-\frac{1}{19^7}-\frac{1}{23^7}+\frac{1}{25^7}+\frac{1}{29^7}\cdots\\&=\frac{\pi}{3}\\
&1+\frac{1}{2^7}-\frac{1}{3^7}-\frac{1}{4^7}+\frac{1}{6^7}+\frac{1}{7^7}-\frac{1}{8^7}-\frac{1}{9^7}+\frac{1}{11^7}+\frac{1}{12^7}\cdots\\&=\frac{\sqrt{10+2\sqrt{5}}}{5\sqrt{5}}\pi\\
&1+\frac{1}{2^7}+\frac{1}{3^7}-\frac{1}{4^7}+\frac{1}{6^7}-\frac{1}{7^7}+\frac{1}{8^7}-\frac{1}{9^7}+\frac{1}{11^7}-\frac{1}{12^7}\cdots\\&=\frac{\sqrt{10-2\sqrt{5}}}{5\sqrt{5}}\pi\\
&1+\frac{1}{3^7}-\frac{1}{7^7}-\frac{1}{9^7}+\frac{1}{11^7}+\frac{1}{13^7}-\frac{1}{17^7}-\frac{1}{19^7}+\frac{1}{21^7}+\frac{1}{23^7}\cdots\\&=\frac{\sqrt{10+2\sqrt{5}}}{10}\pi\\
&1+\frac{1}{3^7}+\frac{1}{7^7}-\frac{1}{9^7}+\frac{1}{11^7}-\frac{1}{13^7}+\frac{1}{17^7}-\frac{1}{19^7}+\frac{1}{21^7}-\frac{1}{23^7}\cdots\\&=\frac{\sqrt{10-2\sqrt{5}}}{10}\pi\\
&1+\frac{1}{3^7}+\frac{1}{5^7}+\frac{1}{7^7}-\frac{1}{9^7}-\frac{1}{11^7}-\frac{1}{13^7}-\frac{1}{15^7}+\frac{1}{17^7}+\frac{1}{19^7}\cdots\\&=\frac{\sqrt{2-\sqrt{2}}}{4}\pi\\
&1+\frac{1}{3^7}+\frac{1}{5^7}-\frac{1}{7^7}+\frac{1}{9^7}-\frac{1}{11^7}-\frac{1}{13^7}-\frac{1}{15^7}+\frac{1}{17^7}+\frac{1}{19^7}\cdots\\&=\frac{\sqrt{4-2\sqrt{2}}+\sqrt{2}+1}{8}\pi\\
&1+\frac{1}{3^7}-\frac{1}{5^7}+\frac{1}{7^7}-\frac{1}{9^7}+\frac{1}{11^7}-\frac{1}{13^7}-\frac{1}{15^7}+\frac{1}{17^7}+\frac{1}{19^7}\cdots\\&=\frac{\sqrt{4+2\sqrt{2}}+\sqrt{2}-1}{8}\pi\\
&1+\frac{1}{3^7}+\frac{1}{5^7}+\frac{1}{7^7}-\frac{1}{9^7}-\frac{1}{11^7}+\frac{1}{13^7}-\frac{1}{15^7}+\frac{1}{17^7}-\frac{1}{19^7}\cdots\\&=\frac{\sqrt{4+2\sqrt{2}}-\sqrt{2}+1}{8}\pi\\
&1+\frac{1}{3^7}-\frac{1}{5^7}-\frac{1}{7^7}+\frac{1}{9^7}+\frac{1}{11^7}+\frac{1}{13^7}-\frac{1}{15^7}+\frac{1}{17^7}-\frac{1}{19^7}\cdots\\&=\frac{\sqrt{2}+1+\sqrt{4-2\sqrt{2}}}{8}\pi\\
&1+\frac{1}{3^7}-\frac{1}{5^7}+\frac{1}{7^7}-\frac{1}{9^7}+\frac{1}{11^7}+\frac{1}{13^7}-\frac{1}{15^7}+\frac{1}{17^7}-\frac{1}{19^7}\cdots\\&=\frac{\sqrt{2-\sqrt{2}}}{4}\pi\\
&1+\frac{1}{5^7}+\frac{1}{7^7}+\frac{1}{11^7}-\frac{1}{13^7}-\frac{1}{17^7}-\frac{1}{19^7}-\frac{1}{23^7}+\frac{1}{25^7}+\frac{1}{29^7}\cdots\\&=\frac{\sqrt{6}}{6}\pi\\
&1+\frac{1}{5^7}+\frac{1}{7^7}-\frac{1}{11^7}+\frac{1}{13^7}-\frac{1}{17^7}-\frac{1}{19^7}-\frac{1}{23^7}+\frac{1}{25^7}+\frac{1}{29^7}\cdots\\&=\frac{\sqrt{6}+\sqrt{3}-\sqrt{2}+2}{12}\pi\\
&1+\frac{1}{5^7}-\frac{1}{7^7}+\frac{1}{11^7}-\frac{1}{13^7}+\frac{1}{17^7}-\frac{1}{19^7}-\frac{1}{23^7}+\frac{1}{25^7}+\frac{1}{29^7}\cdots\\&=\frac{\sqrt{6}-\sqrt{3}+\sqrt{2}+2}{12}\pi\\
&1+\frac{1}{5^7}-\frac{1}{7^7}-\frac{1}{11^7}+\frac{1}{13^7}+\frac{1}{17^7}+\frac{1}{19^7}-\frac{1}{23^7}+\frac{1}{25^7}-\frac{1}{29^7}\cdots\\&=\frac{\sqrt{3}+\sqrt{2}+2-\sqrt{6}}{12}\pi\\
&1+\frac{1}{5^7}-\frac{1}{7^7}+\frac{1}{11^7}-\frac{1}{13^7}+\frac{1}{17^7}+\frac{1}{19^7}-\frac{1}{23^7}+\frac{1}{25^7}-\frac{1}{29^7}\cdots\\&=\frac{\sqrt{2}}{6}\pi\\
&1+\frac{1}{5^7}+\frac{1}{7^7}+\frac{1}{11^7}-\frac{1}{13^7}-\frac{1}{17^7}+\frac{1}{19^7}-\frac{1}{23^7}+\frac{1}{25^7}-\frac{1}{29^7}\cdots\\&=\frac{\sqrt{6}+\sqrt{3}+\sqrt{2}-2}{12}\pi\\
\end{align}

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