Let $u(x)$ be a real function with the properties
- $u(x)$ is continuous and non-decreasing in $x$
- $u'(x)$ is non-increasing in $x$
- $u(0)=0$
- $u'(0)=1$.
In other words, $u(x)$ is a utility function.
In a rather old insurance paper the series on the left hand side of $\sum_{k=0}^{\infty} \frac{\lambda^k}{k!}u(-\lambda u(-z)-zk)=0,$ where $z>0$ and $\lambda>0$, is differentiated with respect to $\lambda$ rather non-chalantly and without further comment by differentiation of every summand, i.e differentiation and summation are exchanged. Is the uniform convergence of the series consisting of the derivatives of the summands, i.e $\sum_{k=0}^{\infty} \left(\frac{\lambda^{k-1}}{(k-1)!}u(-\lambda u(-z)-zk)-\frac{\lambda^k}{k!}u'(-\lambda u(-z)-zk)u(-z)\right),$ needed for this exchange to work really that easy to see? To be more specific, how would such a general series be approached?