Coincidence that series of arctan is alternating series of artanh? I noticed that the power series for $\arctan$ is the alternating series of that for $\operatorname{arctanh}$.
Does it have a special meaning or even some kind of special importance?

The hyperbolic and trigonometric functions are real/imaginary counterparts of each other.
$$\arctan(ix)=i\,\text{artanh}(x),\\
\text{artanh}(ix)=i\,\arctan(x).$$
For an odd series,
$$\sum_k a_k(ix)^{2k+1}=i\sum_k a_k(-1)^kx^{2k+1}.$$
So yes, there is a fundamental relation.
