\begin{definition}[Definition:Bernoulli Numbers/Archaic Form/Definition 1]
The old form Bernoulli numbers ${B_n}^*$ are a sequence of rational numbers defined by the exponential generating function:
{{begin-eqn}}
{{eqn | l = \frac x {e^x - 1}
      | r = 1 - \frac x 2 + \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {B_n^* x^{2 n} } {\paren {2 n}!}
      | c = 
}}
{{eqn | r = 1 - \frac x 2 + \frac {B_1^* x^2} {2!} - \frac {B_2^* x^4} {4!} + \frac {B_3^* x^6} {6!} - \cdots
      | c = 
}}
{{end-eqn}}
for $x \in \R$ such that $\size x < 2 \pi$
\end{definition}