## ACMS Conference Proceedings 2009

#### Title

Approvimating Sums of Infinite Series

#### Document Type

Conference Proceeding

5-29-2009

#### Abstract

The Euler-Maclaurin summation formula is frequently used to efficiently estimate sums of infinite series of the form $\sum_{j=1}^{\infty}f(j)$. The purpose of this article is to describe a modification of this numerical technique designed to simplify and reduce the computational effort required to obtain an acceptable estimate of the sum. The modified formula is obtained by replacing $f\left( x\right)$ with an easily constructed polynomial like interpolating function $a\left( x\right)$ designed to simplify the calculation of the integral and derivatives associated with Euler-Maclaurin. This approach provides a more tractable algorithm which can be written as a matrix equation. Examples are provided to demonstrate that the accuracy of the new algorithm compares favorably with that of the traditional formula. The paper concludes with a brief discussion of a method for approximating the error incurred when replacing the exact value of the sum of the original series with the estimate.

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