Equivalence of Definitions of Bernoulli Numbers

Theorem
The following definitions of the Bernoulli Numbers are equivalent:

Generating Function
The Bernoulli Numbers are defined by the exponential generating function:


 * $\displaystyle \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac{B_n x^n} {n!}$

Recurrence Relation
The Bernoulli Numbers are defined by the recurrence relation:


 * $B_0 = 1; \ \displaystyle B_n = - \sum_{k \mathop = 0}^{n-1} \binom {n} {k} \frac {B_k} {n - k + 1}$

Proof
Starting with the Generating Function definition:

By equating coefficients, we can deduce:


 * $B_0 = 1$ and $\displaystyle \sum_{k \mathop = 0}^n \binom n k \frac {B_k} {n - k + 1} = 0$

Solving this relation, we have:


 * $B_0 = 1; \ \displaystyle B_n = - \sum_{k \mathop = 0}^{n - 1} \binom n k \frac {B_k} {n - k + 1}$