  The theory of summability of divergent series is a major branch of
mathematical analysis that has found important applications in engineering and
science. It addresses methods of assigning natural values to divergent sums,
whose prototypical examples include the Abel summation method, the Cesaro
means, and Borel summability method. In this paper, we introduce a new
summability method for divergent series, and derive an asymptotic expression to
its error term. We show that it is both regular and linear, and that it arises
quite naturally in the study of local polynomial approximations of analytic
functions. Because the proposed summability method is conceptually simple and
can be implemented in a few lines of code, it can be quite useful in practice
for computing the values of divergent sums.
