I want to find the inverse function of the power series,
$$ f(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n+1)!} $$
The only think I can think of that could possibly help is that
$$ f(x)=\frac{\text{sinh}(x)}{x} $$
Also, I've found the first few terms using by calculating the first couple derivatives of $f^{-1}$ at $x=1.1$ (there's a vertical tangent at $x=1$, and $f^{-1}$ is only defined for $[1,\infty)$) but I feel like there must be a better way...
In case you're curious, this power series arises naturally in the following scenario. Say that you hang a wire from two points at equal heights. The wire follows the path
$$ y(x)=k\text{cosh}(x/k) $$
where $k$ is some unknown parameter. I want to find an expression for $k$ in terms of $d$, the distance between the two points, and $L$, the length of the wire. After doing an arc-length integral, a Taylor expansion, and some algebra, I arrive at the equation
$$ \frac{L}{d}=f(d/2k) $$
which, once inverted and rearranged, we get
$$ k=\frac{d/2}{f^{-1}(L/d)} $$