On the transformations of the logarithmic series

Abstract

In this paper we consider transformations of the series \[ l(x) = \sum_{n=1}^{\infty} \frac{x^n}{n}\hspace{5mm} and\hspace{3mm} L(z) = \sum_{n=0}^{\infty}\frac{z^{2n+1}}{2n+1} \] in the forms: (A) $l(x)=\sum_{n=1}^{\infty}\frac{A_{n}x^n}{1-\alpha_nx}$, (B) $L(z)=\sum_{n=0}^{\infty}\frac{B_n}{1-b_nz^2}\bigg(\frac{z}{1-\beta_nz^2}\bigg)^{4n+1}$ and (C) $l(x)=\sum_{n=1}^{\infty}\frac{C_nx^n}{(1-\gamma_1x)...(1-\gamma_nx)}$. Minimization of the coefficients in (A) and (B), under the restrictions $|\alpha_n|,|\beta_n|\leq1$, is explored numerically. The resulting hypothesis is that we can accelerate the convergence like a geometric progression. We prove that the unique lacunary series $l(x)=\sum_{i=0}^{\infty}\frac{A_ix^{2i+1}}{1-\alpha_ix}$ and $L(z)=\sum_{i=0}^{\infty}\frac{B_iz^{4i+1}}{1-b_iz^2}$ diverge for $x \neq 0$ and $z \neq 0$. Assuming $|\gamma_n| \leq 1$ we prove lower and upper bounds for the optimal rate of convergence of (C). A similar upper bound for (A) is proved. Also, some new accelerated series for the logarithmic and other transcendental functions are obtained.