Hyperbolic tangent
The power series for the hyperbolische tangent is written as
Explanation
The derivatives are
⁽⁵⁾ = 16 tanh x − 136 tanh³ x.gif){kind=small}
The point x = 0 gives tanh (0) = 0 so
We substitute this in the Taylor series and find
so that
Example 1
You can see that tanh (0) = 0, because
tanh (0) = 0 − 0 + 0 − 0 + ⋯ = 0