  In the sixth chapter of his notebooks Ramanujan introduced a method of
summing divergent series which assigns to the series the value of the
associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin
summation formula to the partial sums of the series. This method is now called
the Ramanujan summation process. In this paper we calculate the Ramanujan sum
of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and
$\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$.
We find a surprising relation between the two sums when $j=1$ from which
follows a formula that connects the derivatives of the Riemann zeta - function
at the negative integers to the Ramanujan summation of the divergent Euler sums
$\sum_{n\ge 1} n^kH_n, k \ge 0$, where $H_n= H_n^{(1)}$. Further, we express
our results on the Ramanujan summation in terms of the classical summation
process called the Borel sum.
