After I've read Kummer's recipe to get the acceleration of a series, I want to do an example related with the Möbius function $\mu(n)$.
Question. I present my calculations, please A) I would like to know if there are mistakes in how I've used Kummer's acceleration method, specially I need to show that (being required by the method) $$s=\sum_{k=0}^\infty\frac{1}{(k+1)^2}\sum_{j=1}^{k+1}\frac{\mu(j)}{j^3}$$ is convergent; and B) please I would like if there are some method, theoretic or well numeric to get an idea about how was accelerated the series, I say a comparison between the old series $s$, and the new expression $$s=\frac{1}{\zeta(3)}\frac{\pi^2}{6}+\sum_{k=0}^\infty\left( \left( 1-\frac{1/\zeta(3)}{\sum_{j=1}^{k+1}\mu(j)/j^3} \right) \frac{\sum_{j=1}^{k+1}\frac{\mu(j)}{j^3}}{(k+1)^2} \right) .$$ Many thanks.
The calculations that I did to get A) was the specialization of Kummer's method that is explained in this [1], with $a_k=\frac{1}{(k+1)^2}\sum_{j=1}^{k+1}\frac{\mu(j)}{j^3}$, and $c_k=\frac{1}{(k+1)^2}$. From particular values of Dirichlet series one has $$\lim_{k\to\infty}\sum_{j=1}^{k+1}\frac{\mu(j)}{j^3}=\frac{1}{\zeta(3)},$$ and from the method $$s=\frac{1}{\zeta(3)}\frac{\pi^2}{6}+\sum_{k=0}^\infty\left( \left( 1-\frac{1/\zeta(3)}{\sum_{j=1}^{k+1}\mu(j)/j^3} \right) \frac{\sum_{j=1}^{k+1}\frac{\mu(j)}{j^3}}{(k+1)^2} \right). $$ But I need to show previously convergence for the genuine $s$. I try do it using that the convergence absolute of a series implies its convergence. Thus my calculations were using Apostol's Theorem 3.2 to show from $$ \left| \sum_{k=0}^\infty\frac{1}{(k+1)^2}\sum_{j=1}^{k+1}\frac{\mu(j)}{j^3}\right|<\sum_{k=0} ^\infty\frac{1}{(k+1)^2} \left( \zeta(3) -\frac{1}{(k+1)^2}+O \left( \frac{1}{(k+1)^3} \right) \right), $$ that the genuine $s$ is convergent because previous RHS is $\frac{\zeta(3)}{\zeta(2)}-\frac{\zeta(4)}{2}+\sum_{k=0}^\infty\frac{1}{(k+1)^2}O\left( \frac{1}{(k+1)^3} \right)$ and thus $O(1)$. In this last step is where I've doubts about how write a rigurous proof of the convergence of the genuine series $s$. And what I am asking in B) us how to get an idea of the accuracy of the digits from the acceleration that we did, I am saying a comparison of accuracy of the digits of both series, well numeric providing a clai about how more or less is improved the convergence of the seies, or well with a theoretic reasoning.
References:
[1] Convergence Improvement, MathWorld.
[2] Apostol, An Introduction to Analytic Number Theory, Springer (1976).