ネイピア数の表現
{\displaystyle e=\sum _{k=0}^{\infty }{\frac {(3k)^{2}+1}{(3k)!}}}
e = [ ∑ k = 0 ∞ 4 k + 3 2 2 k + 1 ( 2 k + 1 ) ! ] 2 {\displaystyle e=\left[\sum _{k=0}^{\infty }{\frac {4k+3}{2^{2k+1}\,(2k+1)!}}\right]^{2}}
e = − 12 π 2 [ ∑ k = 1 ∞ 1 k 2 cos  ( 9 k π + k 2 π 2 − 9 ) ] − 1 / 3 {\displaystyle e=-{\frac {12}{\pi ^{2}}}\left[\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\ \cos \left({\frac {9}{k\pi +{\sqrt {k^{2}\pi ^{2}-9}}}}\right)\right]^{-1/3}}
e = ∑ k = 1 ∞ k 2 2 ( k ! ) {\displaystyle e=\sum _{k=1}^{\infty }{\frac {k^{2}}{2(k!)}}}
e = ∑ k = 1 ∞ k 3 5 ( k ! ) {\displaystyle e=\sum _{k=1}^{\infty }{\frac {k^{3}}{5(k!)}}}
e = ∑ k = 1 ∞ k n B n ( k ! ) {\displaystyle e=\sum _{k=1}^{\infty }{\frac {k^{n}}{B_{n}(k!)}}} (Bn は n 番目のベル数)