rwg>I like to present convergent telescoping identities as infinite sums, e.g.,
... But this example immediately suggests
k - 1 ==== x \ x cot(--) - cot(x) = > csc(--), k / n 2 ==== 2 n = 0
which blows up in one direction and goes nuts in the other. Duh, inf ==== 2 \ n ------ = > csch(2 x), x / e - 1 ==== n = 0
pretty much the log derivative of Joerg's observation. Test: apply_nouns(subst([x=.69105d0,inf=5],%)) 2.00841519141928d0 = 2.00841519141928d0 (With quadratic convergence, infinity = 5.) jj>[...] f@@n = the iteration for sqrt(1) of order 2^n Note that if we follow each Newton step with a true sqrt, 1 a + ------ n - 1 a n - 1 a = sqrt(---------------), n 2 with a a = sqrt(-), we get 0 b agm(a, b) a a a . . . = ---------. 0 1 2 b --rwg GRABBIEST BAGBITERS (Peter Samson's ancient PDP-1 music compiler used the error code agm to signify: Argument Greedily Masticates.)