Let \(X\) be a random variable with moment generating function \(m(u)\) and cumulant generating function \(c(u)\). Assuming that both functions exist, prove that

\[\left.\frac{d^{2}}{d u^{2}} \frac{m(u+\lambda)}{m(\lambda)}ight|_{t=0}=c^{\prime \prime}(\lambda)\]