I'm doing some research on estimating errors, and the expression $$S(x)=\sum_{i=0}^k\frac{1}{x^{1/2^i}}$$ seems to be coming up every now and then, where $k=\lceil \log_2\log_2(x)\rceil$. I'm wondering what its limit is as $x\to \infty$. Since the size of the sum depends on $x$, it makes it a little difficult. It doesn't seem to be converging to $0$ since the last term is always between $1/2$ and $1$.
I imagine this sum is related to the sum $$T(x)=\sum_{i=0}^k x^{2^i}$$The relationship can be seen as $T(x)=S(1/x^{2^k})$. Of course, $T(x)$ is only defined for $x>1$ while $S(x)$ is defined for arbitrarily large $x$.