How to show that this series of rational functions has a maximum at $x=0$ using the “Descartesschen Regel”?

Question

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'' (Descartes rule).

What is the Descartes rule that is mentioned here? Can the maximum at $x = 0$ be shown using an elementary method as the paper claims?

Answer 1

The function $f$ does not even have a local maximum at $0$.

Indeed, using partial fraction decomposition, it is easy to see that $$f(x)=\frac{x+1}{x+2}-(x+1)^2 (\psi'(x+2)+(x+1) \psi''(x+2)), $$ where $\psi=(\ln\Gamma)'$, the digamma function, so that $$\psi'(x)=\sum_{k=0}^\infty\frac1{(x+k)^2}\quad\text{and}\quad \psi''(x)=-2\sum_{k=0}^\infty\frac1{(x+k)^3} $$ -- see e.g. Andrews et al., formula (1.2.14), page 13. Hence, $$f'(0)=8 \zeta (3)+\frac{1}{4}-\frac{1}{15} \pi ^2 \left(5+\pi ^2\right) =0.082647\ldots>0. $$ Here is the graph of $f$, which suggests that $f$ is actually increasing on $[0,1]$:

Answer 2

I can give an answer to your first question - below, I'll give a translation of the "Descartessche Regel" from a suitably old German book (to make the experience authentic).

Before getting to that, though, I wonder: Are you sure your paper is claiming a maximum of $f$ itself at $x=0$, and not just some part of the expression, such as 1/denominator? I ask not only in view of Iosif Pinelis's observation, but because the Descartessche Regel is a statement about polynomials. If in doubt, by all means post the German text and I can provide a translation.

So, from "Weber-Wellstein Enzyklop\"adie der Elementarmathematik, Erster Band: Arithmetik, Algebra und Analysis", by Heinrich Weber, updated by Paul Epstein, 4th edition, Teubner, 1922:

The number of positive roots [of a polynomial with real coefficients with nonzero constant term] is at most equal to, or less by an even number than, the number of sign changes in the coefficients [counting in order of decreasing powers]. Multiple roots are counted according to their multiplicity.