Questions on Bernoulli numbers, a special sequence of rational numbers that arise as the coefficients in the power series expansions of certain elementary functions.
The $n$th Bernoulli number $B_n$ is frequently defined in terms of a generating function:
$$\frac x{1-e^{-x}}=\sum\limits_{n = 0}^\infty B_n\frac{x^n}{n!}$$
The first few Bernoulli numbers are
\begin{align*} B_0 &=1 \\ B_1 &=\frac12 \\ B_2 &=\frac16 \\ B_3 &=0 \\ B_4 &=-\frac1{30} \\ B_5 &=0 \end{align*}
All Bernoulli numbers with $n$ odd, except for $B_1$, are zero.
Alternatively, the $n$th Bernoulli number is the constant coefficient in the $n$th Bernoulli polynomial $B_n(x)$, which can be defined in terms of a generating function as well:
$$\frac{te^{-xt}}{1-e^{-t}} = \sum_{k=0}^\infty B_n(x)\frac{t^n}{t!}$$
The Bernoulli numbers have deep connections to number theory, and frequently rise in combinatorics and asymptotic estimates of functions, as well.
