m

n

$-\pi B_1\left(\frac{1}{n}\right)={\displaystyle {\sum }_{k>0}\frac{1}{k}\sin \left(\frac{2\pi k}{n}\right)}$

1

1

$-\pi B_1\left(1\right)-\frac{1}{2}\pi =\left(\frac{0}{1}+\frac{0}{2}+\frac{0}{3}+\frac{0}{4}+\cdots \right)$

2

$0=\left(\frac{0}{1}-\frac{0}{2}+\frac{0}{3}-\frac{0}{4}+\cdots \right)$

3

$-\pi B_1\left(\frac{1}{3}\right)=\frac{\sqrt{3}}{2}\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{4}+\cdots \right)$

4

$-\pi B_1\left(\frac{1}{4}\right)=\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots \right)$

Gregory-Leibnitz formula

6

$-\pi B_1\left(\frac{1}{6}\right)=\frac{\sqrt{3}}{2}\left(\frac{1}{1}+\frac{1}{2}-\frac{1}{4}-\frac{1}{6}+\frac{1}{7}+\cdots \right)$

8

$\left(\frac{\sqrt{2}}{2}\frac{1}{1}+\frac{1}{2}+\frac{\sqrt{2}}{2}\frac{1}{3}-\frac{\sqrt{2}}{2}\frac{1}{5}-\frac{1}{6}-\frac{\sqrt{2}}{2}\frac{1}{7}+\frac{1}{9}\frac{\sqrt{2}}{2}+\cdots \right)$

Newton-Euler formula

12

$\begin{array}{l}-\pi B_1\left(\frac{1}{12}\right)=\left(\frac{1}{2}\frac{1}{1}+\frac{\sqrt{3}}{2}\frac{1}{2}+\frac{1}{3}+\frac{\sqrt{3}}{2}\frac{1}{4}-\frac{1}{2}\frac{1}{5}\right)\\ -\frac{1}{2}\frac{1}{7}-\frac{\sqrt{3}}{2}\frac{1}{8}-\frac{1}{9}-\frac{\sqrt{3}}{2}\frac{1}{10}-\frac{1}{2}\frac{1}{11}+\cdots \end{array}$

m

n

$-\pi B_2\left(\frac{1}{n}\right)={\displaystyle {\sum }_{k>0}\frac{1}{k^2}\sin \left(\frac{2\pi k}{n}\right)}$

2

1

${\pi }^2\frac{1}{6}=\left(\frac{0}{1^2}+\frac{0}{2^2}+\frac{0}{3^2}+\cdots \right)$

2

${\pi }^2\frac{1}{12}=\left(\frac{0}{1^2}-\frac{0}{2^2}+\frac{0}{3^2}-\frac{0}{4^2}+\cdots \right)$

3

$\pi B_2\left(\frac{1}{3}\right)=\frac{\sqrt{3}}{2}\left(\frac{1}{1^2}+\frac{1}{2^2}-\frac{1}{4^2}-\frac{1}{5^2}+\frac{1}{7^2}+\cdots \right)$

4

$\pi B_2\left(\frac{1}{4}\right)=\left(\frac{1}{1^2}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\cdots \right)$

6

$\pi B_2\left(\frac{1}{6}\right)=\frac{\sqrt{3}}{2}\left(\frac{1}{1^2}+\frac{1}{2^2}-\frac{1}{4^2}-\frac{1}{6^2}+\frac{1}{7^2}+\cdots \right)$

8

$\begin{array}{l}\pi B_2\left(\frac{1}{8}\right)=(\frac{\sqrt{2}}{2}\frac{1}{1^2}+\frac{1}{2^2}+\frac{\sqrt{2}}{2}\frac{1}{3^2}-\frac{\sqrt{2}}{2}\frac{1}{5^2}-\frac{1}{6^2}\\ -\frac{\sqrt{2}}{2}\frac{1}{7^2}+\frac{1}{9^2}\frac{\sqrt{2}}{2}+\cdots )\end{array}$

12

$\frac{1}{2}\left(\frac{1}{1^2}+\sqrt{3}\frac{1}{2^2}+2\frac{1}{3^2}+\sqrt{3}\frac{1}{4^2}+\frac{1}{5^2}\right)$

$\frac{1}{2}\left(-\sqrt{3}\frac{1}{7^2}-\frac{1}{8^2}-2\frac{1}{9^2}-\sqrt{3}\frac{1}{{10}^2}-\frac{1}{{11}^2}\right)+\cdots$