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

[robve]

(02-14-2021 11:30 AM)Albert Chan Wrote:  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)

Thanks, that explains it. I misunderstood the extrapolation was for S2, not the infinite alternating series (which couldn't be much of an improvement).

Note that in general one should run the summation loop for S2 in Euler_transform backwards to accumulate fewer roundoff errors because the summand terms t[i] diminish quickly. Though I can't see how it really improves the final S that is already an approximation.

- Rob