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

[Albert Chan]

(02-13-2021 11:30 PM)Albert Chan Wrote:  Or, Secant's method, from 2 points: (x1, y1) = (b, t[d]), (x2, y2) = (c, t[d+1])

Selection of points is arbitrary, as long as it is consistent.
Example, we can use mean of 2 cumulative sum for x's, estimated error gap for y's

(x1, y1) = ((b+a)/2, (b-a)/2)
(x2, y2) = ((c+b)/2, (c-b)/2)

Again, Secant's method, extrapolate for (x, 0)

x = (c+b)/2    - (c-b)/2 * ((c+b)/2 - (b+a)/2) / ((c-b)/2 - (b-a)/2)
  = c - (c-b)/2 - (c-b)/2 * (c-a) / (c-2b+a)
  = c - (c-b)/2 * ((c-2b+a) + (c-a)) / (c-2b+a)
  = c - (c-b)^2 / (c-2b+a)
  = Aitken(a, b, c)