Let $a_n = x + x^{\frac{1}{2}} + x^{\frac{1}{4}} + \dots + x^{\frac{1}{2^n}}$ where $x\in \mathbb{R}$ and $x\gt0$. Find a closed formula for $a_n$ in terms of $x$ or at least an approximation. I need this approximation for computational complexity. Actually, big theta $\Theta$ is what I'm looking for. See this: https://en.wikipedia.org/wiki/Big_O_notation
My try: Obviously $a_n$ diverges for all $x\not = 0$. Also the geometric series isn't useful here. If we define sequence recursively by $a_{n+1} = a_n + x^{\frac{1}{2^{n+1}}}$ and $a_0 = x$, it can't be solved still using techniques for recursive sequences and I'm stuck here.
For $x$ close to $1$, $x^{1/2^n}\approx 1+\dfrac{x-1}{2^n}$ and the sum of the $n$ first terms is about $n+x-1$.
For large $x$, after sufficiently many square roots, the value comes close to $1$.
An approximation is for instance
$$x+\sqrt[2]x+\sqrt[4]x+\sqrt[8]x+\sqrt[16]x+n-4+\sqrt[16]x-1.$$