Closed form for $\sum_{0\le x\lt\infty}n^{2^{-x}}-1$ Is there a closed form for the expression below?
$$f(n)=\sum_{0\le x\lt\infty}n^{2^{-x}}-1=(n-1)+(\sqrt{n}-1)+(\sqrt{\sqrt{n}}-1)+\cdots$$
Approximations are good as well. This appeared to me while analyzing an algorithm.

After some analysis I was unable to find a closed form, but here are some good approximations:
On $n\in(0,7)$, $f(n)$ is close to the form of $\alpha + \beta \sqrt n + \gamma n$, and for $n\in[7,\infty)$, is of the form $\lambda + \rho n + \chi \log n$. So we can approximate with,
$$ f(n) \approx \begin{cases}
 \alpha + \beta \sqrt n + \gamma n, & n\in(0,7) \\
 \lambda + \rho n + \chi \log n, & n\in[7,\infty) \\
\end{cases} $$
You can use any fitting algorithm to approximate these values. I came up with,
\begin{align*}
 \alpha &= -3.03894 &\lambda &= -3.23294 \\
 \beta &= 2.17427  &\rho &= 1.04346 \\
 \gamma &= 0.868928  &\chi &= 2.28405 \\
\end{align*}
If computing a logarithm is too expensive or computation heavy, note that $f$ is approximately linear as $n$ grows. You can approximate $f(n)\approx 1.09352 n + 2.82223$.
