using (5/2)x - (5/6)x^3 + ...

[tommy1729]

(5/2) x - (9/8 ) x^2 + ...

has a solution 

exp(x) -  exp(-x) + exp(-2x) - exp(- 5/2 x)

To infinity 

lim (exp(x) -  exp(-x) + exp(-2x) - exp(- 5/2 x))^2 - (2 sinh(x))^2  = 0

lim (exp(x) -  exp(-x) + exp(-2x) - exp(- 5/2 x))^3 - (2 sinh(x))^3  = 3


This behaves nicer for negative x too.

exp(x) -  exp(-x) + exp(-2x) - exp(- 5/2 x) is the new favorite for now.

regards

tommy1729