A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine (Q6611509)

The authors pose themselves two problems:\N\N\textbf{1.} What is the MacLaurin expansionn of the even function\N\[\NF(x)=\begin{cases} \ln{\frac{2(1-\cos{x})}{x^2}}, &0<|x|< 2 \pi \\\N0, &x=0 \end{cases}\N\]\N\N\textbf{2.} Is the even function\N\[\NR(x)=\begin{cases} \frac{\ln{\frac{2(1-\cos{x})}{x^2}}}{\ln{\cos{x}}}, &0<|x|<\frac{\pi}{2} \\\N\frac{1}{6}, &x=0 \\\N0, &x=\pm \frac{\pi}{2} \end{cases}\N\]\Ndecreasing on the interval, $[0,\frac{\pi}{2}]$?\N\NThe answers are given in the paper and will be indicated below.\N\NThe layout of the paper is as follows\N\N\textbf{\S1. Motivation} ($1$ page)\N\N\textbf{\S2. Lemmas} ($1\frac{1}{2}$ page)\N\N\textbf{\S3. MacLaurin power series expansion} ($4$ pages)\N\NThe series is given (\textbf{Theorem 1}) in the form\N\[\NF(x)=-\sum_{n=1}^{\infty}\,\frac{E_{2n}}{(2n)!},\ |x|<2\pi,\N\]\Nwhere the numbers $E_{2n}$ are given explicitly as the values of intricate determnants.\N\N\textbf{\S4. Decreasing property} ($4\frac{1}{2}$ pages)\N\NThis is \textbf{Theorem 2}:\N\NThe function $R(x)$ is decreasingly maps $[0,\frac{\pi}{2}]$ onto $[0,\frac{1}{6}]$.\N\N\textbf{\S5. Conclusion} ($\frac{1}{2}$ page)\N\N\textbf{References} (16 items)