Taylor series
Exponential function

The exponential function e x {\displaystyle e^{x}} (with base e) has Maclaurin series

e x = ∑ n = 0 ∞ x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + ⋯. {\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots.} It converges for all x.

The exponential generating function of the Bell numbers is the exponential function of the predecessor of the exponential function:

exp (exp x − 1) = ∑ n = 0 ∞ B n n ! x n {\displaystyle \exp(\exp {x}-1)=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}x^{n}}
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