Do any of the standard methods of acceleration convergence of series, when applied to
the series $1 - 1 + 1/2 - 1/2 + 1/3 - 1/3 + ...$, give convergence to 0 with error $o(1/n)$?


I tried applying Euler's method to the series, and found that the estimates fall like $1/n$; in fact, the $n$th estimate seems to be to $2/n + o(1/n)$ (e.g., for $n=500$, the $n$th Euler's method estimate is about 1.992 over 500). There are better methods of accelerating convergence, but I don't know much about them.


More broadly, I am looking for a method of accelerating convergence of sequences in which the $n$th term is the mean of the first $n$ terms of a bounded almost periodic sequence. (E.g., in the case where the almost periodic sequence is the periodic sequence $1,-1,1,-1,1,-1,...$, the sequence of averages is $1/1,0/2,1/3,0/4,1/5,0/6,...$, which corresponds to the series $1/1-1/1+1/3-1/3+1/5-1/5+...$.)



accelerating convergence of a class of sequences
The Aitken delta squared method gives $O(1/n^2)$.


