I am reading an old German paper, and at one step they mention that the function \begin{equation*} f(x) := \sum_{k=2}^\infty \frac{(1+x)(k(k-1)^2 + (2+x)(1+x)^2)}{(k+x)^3 (k + 1 + x)^2} \end{equation*} defined on $x \in [0, 1]$ can be shown to have a maximum at $x = 0$ using an elementary application of the ''Descartesschen Regel'' (Descarte rule).
What is the Descarte rule that is mentioned here? Can the maximum at $x = 0$ be shown using an elementary method as the paper claime?