I am trying to numerically sum a series $a_n$ which converges slowly. Although a bound on the remainder $\sum_{k>N} a_k$ can be computed fast. To find an estimate of the sum $S = \sum_{k>=1} a_k$, I am using Aitken's method which transforms the series $\{a_n\}$ to $\{b_n\}$ such that $\sum_{k>=1} b_k = S$ and $\{b_n\}$ converges fast. However, I cannot find a tight bound on the remainder $\sum_{k>N} b_k$ that can be computed fast. Since I am uninitiated to numerical analysis in general, any pointer would be helpful.
The terms computed by Aitken's method are ratios and it's hard to compute a tight bound on them.
