アペリーの定数
ζ ( 3 ) = 7 180 π 3 − 2 ∑ n = 1 ∞ 1 n 3 ( e 2 π n − 1 ) {\displaystyle \zeta (3)={\frac {7}{180}}\pi ^{3}-2\sum _{n=1}^{\infty }{\frac {1}{n^{3}(e^{2\pi n}-1)}}}
ζ ( 3 ) = 14 ∑ n = 1 ∞ 1 n 3 sinh  ( π n ) − 11 2 ∑ n = 1 ∞ 1 n 3 ( e 2 π n − 1 ) − 7 2 ∑ n = 1 ∞ 1 n 3 ( e 2 π n + 1 ) {\displaystyle \zeta (3)=14\sum _{n=1}^{\infty }{\frac {1}{n^{3}\sinh(\pi n)}}-{\frac {11}{2}}\sum _{n=1}^{\infty }{\frac {1}{n^{3}(e^{2\pi n}-1)}}-{\frac {7}{2}}\sum _{n=1}^{\infty }{\frac {1}{n^{3}(e^{2\pi n}+1)}}}