On the alternating series \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots\)
The remainder \(R_n:=\sum^\infty_{k=n+1} (-1)^{k+1} k^{-1}\) is provided best estimates \(a_n\leq|R_n|<b_n\), \(n\geq 1\). From what is known for alternating series in general, \(|R_n|< (n+1)^{-1}\). The first author had sharpened this by \(|R_n|< (2n+1)^{-1}\) and had given some estimate \((2n+\alpha)^{-1} <|R_n|\), in [Rev. Anal. Numér. Théor. Approx. 21, 195-199 (1992; Zbl 0801.40004)]. Now it is shown that, in fact, \((2n+1)^{-1} =b_n\) and that \(a_n=(2n+a)^{-1}\) with \(a=x_1\) from \((2n+x_n)^{-1}=|R_n|\), that is \(a=(1-\log 2)^{-1}-2\). For the proofs, \((x_n)\) is examined.
scientific article; zbMATH DE number 1956899