(34C) (11C) Summation of Infinite Alternating Series

[robve]

I have worked through this nice paper and algorithm too, many thanks Valentin!

(02-12-2021 03:04 PM)Albert Chan Wrote:  We may apply Aitken Extrapolation, to slight improve estimate of s2

lua> s1 - (s2 - t[d+1]^2/(t[d+1] - t[d]))
0.6931471807531758
lua> log(2)
0.6931471805599453

Very clever to think of that. I have one question. I would think that t[d+1]^2/(t[d+1] - t[d]) and (x[n+1]-x[n])^2/((x[n+1]-x[n])-(x[n]-x[n-1])) are related if terms x[] are extrapolated, but that is not the case? Shouldn't the formula be (t[d]+t[d+1])^2/(t[d+1]) since when d=2 for example t[2]=x[n+2]-x[n+1] and t[3]=x[n+3]-2x[n+2]+x[n+1]? How did you derive your formula?

- Rob