# How to find the inverse or a tight bound on a series

If $$f(x)=1-\frac{4}{\pi}\sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1}e^{-\frac{\pi^2 (2k+1)^2}{8 x^2}}$$, find $$\min\{x:f(x)\geq 1-y \}$$.

The function $$f(x)$$ is increasing and its output falls in $$[0,1]$$. Usually, we have to find the inverse of $$f(x)$$. I don't know how to do this. If we don't have an inverse, we should bound $$f(x)$$. Even if we consider the first three terms of $$f(x)$$ to bound it, the inverse of the sum of exponentials is not easy to find, and it is not a good bound for me. Any idea?

I tried $$erf(x)$$ but it is not always was not a valid bound on the whole region of $$x \geq 0$$. $$erf(x)-1$$ is somehow close, but still, some part of the x-axis is lower bound and other parts it is upper bound.

The function $$f(x)$$ is CDF of maximum of absolute value of a standard Weiner process in $$t\in [0,1]$$.