An Euler-Maclaurin Transformation of a Slowly Convergent Series with an Application to Fourier Coefficient Evaluation

A standard procedure in numerical treatment of a slowly convergentseries is to transform it into a new series, convergent or asymptotic,so that far fewer terms of the new series are required to obtainthe desired accuracy. Particularly, when the general term ofthe original series depends on a small-valued parameter , oneis tempted to consider an expansion in powers of , which istantamount to interchanging the order of summation. We considerseries where, as a result of this interchange, almost all theseries formally corresponding to the coefficients of ascendingpowers of diverge. Invoking the Euler-Maclaurin summation formula,it is shown that under certain simple conditions, these divergentseries can be replaced by values obtained from the analyticcontinuation of a Dirichlet series at decreasing values of itsargument. Several examples are discussed, including an applicationto Fourier coefficient evaluation of certain periodic analyticfunctions.