A Challenging Test For Convergence Accelerators: Summation Of A Series With A Special Sign Pattern∗

Slowly convergent series that have special sign patterns have been used in testing the efficiency of convergence acceleration methods. In this paper, we study the series Sm,n = P∞ k=0 (−1) k/m (k+1)2n+1 when m ≥ 1, n ≥ 0, which has m positive terms followed by m negative terms periodically. Using special functions, we first derive its sum in simple terms involving only the Riemann Zeta function and trigonometric functions. With the exact sum available, we next use this series to test the efficiency of various nonlinear convergence acceleration methods in summing it numerically. We conclude that the Shanks transformation and the Levin—Sidi d-transformation are two acceleration methods that produce highly accurate approximations to the sum of Sm,n, the latter being the more effective.