Sum of Bernoulli Numbers by Binomial Coefficients Vanishes

Theorem

 * $\forall n \in \Z_{>1}: \displaystyle \sum_{k \mathop = 0}^{n - 1} \binom n k B_k = 0$

where $B_k$ denotes the $k$th Bernoulli number.

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.

Also see

 * Definition:Bernoulli Numbers/Recurrence Relation