Convergence acceleration of logarithmically convergent series avoiding summation

AbstractQuite often in application, logarithmically convergent series have to be evaluated. There are several convergence acceleration methods that are based on the evaluation of partial sums sn for relatively large n, and thus, normally require the evaluation of all terms aj with 0 ≤ j ≤ n. Here, we show that it is possible to avoid the computation of the partials sums of high order if it is possible to evaluate a few terms aj for relatively large j. The effectiveness of the approach is demonstrated for the 1z expansion that is a particularly difficult example of logarithmic convergence