The exponential generating function for the Bernoulli numbers is

We develop an equation involving sums of Bernoulli numbers on one side, and a simple generating involving powers of $e$ that gives us the appropriate sum of powers on the other side. Equating coefficients of powers of $x$ then gives the result.

To get a generating function where the coefficient of $x^{n}/n!$ is $\sum_{{m=1}}^{{n-1}}m^{k}$, we can use

But this is also a geometric series, so

Equating coefficients of $x^{k}/k!$ we get

which proves the first equality.

If $f(x)$ is a polynomial, write $[x^{r}]f(x)$ for the coefficient of $x^{r}$ in $f(x)$. Then

and thus if $r\leq k$, iterating, we get

Then using the fact that $b_{k}^{{\prime}}=kb_{{k-1}}$, we have

Now reverse the order of summation (i.e. replace $r$ by $k+1-r$) to get