ProofWiki:Sandbox

Proof
Take the definition of Bernoulli numbers:
 * $\displaystyle \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$

From the definition of the exponential function:

Thus:

By Product of Absolutely Convergent Series, we will let:

Then:

$\forall n \in \Z_{\gt 0}$, the sum of the coefficients of $x^n$ in $c_n$ equal $0$.

Multiplying $c_n$ through by $\paren {n + 1 }!$ gives:

But those coefficients are the binomial coefficients:

Hence the result.