HP 42S, DM 42 Integral

[Albert Chan]

(02-13-2021 01:38 AM)Thomas Okken Wrote:  I am a bit surprised because I have observed the death spiral towards zero before ...

It may be hard to see death spiral to 0, with INTEG u-transformation, and romberg extrapolation.
However, if we evaluate integral other ways, it is not hard to show one.

\(\displaystyle AGM(a,b) = {(a+b)(\pi/4) \over K\left(k = {a-b \over a+b}\right)} \)

AGM(a,b) = arithmetic-geometic mean of a, b (above is equation 8)
K = complete elliptic integral of first kind
k = elliptic modulus

(pi/2) / K(1) = AGM(2,0) = AGM(1,0) = AGM(1/2,0) = AGM(1/4,0) = ... = 0

Arithmetic mean forms a death-spiral toward zero.
Thus, it may be better to test geometric mean convergence, AGM(2,0) = AGM(1,0) = 0