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In the sixth chapter of his notebooks Ramanujan introduced a method ofsumming divergent series which assigns to the series the value of theassociated Euler-MacLaurin constant that arises by applying the Euler-MacLaurinsummation formula to the partial sums of the series. This method is now calledthe Ramanujan summation process. In this paper we calculate the Ramanujan sumof 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 whichfollows a formula that connects the derivatives of the Riemann zeta - functionat 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 expressour results on the Ramanujan summation in terms of the classical summationprocess called the Borel sum.

Ramanujan Summation and the Exponential Generating Function $ \sum_{k=0}^{\infty}\frac{z^{k}}{k!}\zeta ^{\prime}(-k)$