On Tue, Jan 2, 2018 at 2:58 PM, françois mendzina essomba2 <m_essob@yahoo.fr
wrote:
hello...
tanh(x) =-csch(2*x)+1/(-csch(4*x)+1/(-csch(8*x)+1/(-csch(16*x)+1/(- csch(32*x)+1/(-csch(64*x)+1/(-csch(128*x)+1/(…)))))));
approximation.
tanh(x)=-csch(2*x)+1/(-csch(4*x)+1/(-csch(8*x)+1/(-csch(16* x)+1/(-csch(32*x)+1/(-csch(64*x)+1/(-csch(128*x)+1/(1)))))));
FME...
Dr. Essomba, perhaps because I am old and rusty, I don't recall ever seeing such a result. The convergence rate of a continued fraction is an increasing function of the (magnitudes of the) denominators, which vanishes along with them. The only way yours makes sense numerically is with that 1 you have tacked onto the tail. The value is chaotically sensitive to replacing that "1" with a variable. Mathematica: Coth@x == ContinuedFractionK[1, -Csch[2^k x], {k, \[Infinity]}] ContinuedFractionK::div: The continued fraction \!\(\*UnderoverscriptBox[\(\[ContinuedFractionK]\), \(k = 1\), \(\[Infinity]\)]\*FractionBox[\(1\), \(-Csch[\*SuperscriptBox[\(2\), \(k\)]\ x]\)]\) does not converge. And yet, with that "1", the "convergence", if you could call it that, is fast and furious. Have you ever seen such continued fractions in the literature? --Bill Gosper