Equivalence of Definitions of Bernoulli Numbers

Theorem
The following definitions of the Bernoulli Numbers are equivalent:

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:


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