On the transformations of the logarithmic series

Abstract

In this paper we consider transformations of the series $$l(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n}andL(z)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^{2n+1}}{2n+1}$$ in the forms: (A) $l(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{A_n{x}^n}{1-{\alpha }_{n}x}$, (B) $L(z)=\sum _{n=0}^{\mathrm{\infty }}\frac{B_n}{1-b_n{z}^2}(\frac{z}{1-{\beta }_n{z}^2}{)}^{4n+1}$ and (C) $l(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{C_n{x}^n}{(1-{\gamma }_{1}x)...(1-{\gamma }_{n}x)}$. Minimization of the coefficients in (A) and (B), under the restrictions $|{\alpha }_n|,|{\beta }_n|\le 1$, is explored numerically. The resulting hypothesis is that we can accelerate the convergence like a geometric progression. We prove that the unique lacunary series $l(x)=\sum _{i=0}^{\mathrm{\infty }}\frac{A_i{x}^{2i+1}}{1-{\alpha }_{i}x}$ and $L(z)=\sum _{i=0}^{\mathrm{\infty }}\frac{B_i{z}^{4i+1}}{1-b_i{z}^2}$ diverge for $x\ne 0$ and $z\ne 0$. Assuming $|{\gamma }_n|\le 1$ 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.