Monotonicity and convexity of the sequences of the areas bounded by the sine and cosine integrals (Q2774479)

After having given the sine and cosine integrals, denoted by si\((x)\) and Ci\((x)\), respectively, it is shown that the respective sequences of the numerical areas bounded by their consecutive zeros are each convex and strictly decreasing to zero. It is emphasized that for the sine integral this concept holds also when the sequence begins with the area between \(x=0\) and the first zero of si\((x)\), whereas for Ci\((x)\) the area between \(x=0\) and the first zero of Ci\((x)\) is less than the area bounded by the next arch. Three theorems and two lemmas structure the article and a conjecture (which concludes the article), which states that the numerical calculations of sequences of areas \(\{|A_k|\}\) and \(\{|B_k|\}\), \(k= 0,1,2,\dots,\) and \(k= 1,2,\dots\), suggest that they form, respectively, completely monotonic sequences.