For large $n$, what is a good upper-bound for $\sqrt{1-x^{1/n}}$ valid for $x \in (0, 1)$?

Question

Let $n$ be a large integer.

Question. What is a good upper-bound for $\sqrt{1-x^{1/n}}$ valid for $x \in (0, 1)$ ?

The trivial bound $\sqrt{1-x^{1/n}} \le \sqrt{(1-x)/n}$ which follows from the (fractional) binomial theorem, is very loose for $0 < x \ll 1$.

Notes

• Philosophically, an upper-bound means to exhibit a relation of the form $f(x) \le g(x)$ where $g(x)$ "looks simpler". For the above question, idealistically, upper-bound with no appearance of $n$ in an exponent of some $x$-expresion...

Edit

Illustrating bound from user Claude's answer Looks pretty tight for large values of $n$, right ?

Answer 1

No $n$ in the approximation seems difficult (at least to me).

Composing Taylor series for large values of $n$ $$\log \left(\sqrt{1-x^{\frac{1}{n}}}\right)=-\frac{1}{2} \log \left({n}\right)+\log \left(\sqrt{-\log (x)}\right)+\frac{\log (x)}{4 n}+O\left(\frac{1}{n^2}\right)$$ giving as a quite good approximation $$\color{blue}{\sqrt{1-x^{\frac{1}{n}}}\sim x^{\frac{1}{4 n}}\sqrt{\frac{-\log (x)}{n}}} $$

Trying for $n=100$ $$\left( \begin{array}{cccc} x & x^{\frac{1}{4 n}}\sqrt{\frac{-\log (x)}{n}} & \sqrt{1-x^{\frac{1}{n}}} & \sqrt{\frac{-\log (x)}{n}}\\ 0.05 & 0.1717904 & 0.1717936 & 0.1730818 \\ 0.10 & 0.1508717 & 0.1508734 & 0.1517427 \\ 0.15 & 0.1370843 & 0.1370853 & 0.1377360 \\ 0.20 & 0.1263542 & 0.1263549 & 0.1268636 \\ 0.25 & 0.1173336 & 0.1173341 & 0.1177410 \\ 0.30 & 0.1093959 & 0.1093963 & 0.1097257 \\ 0.35 & 0.1021923 & 0.1021925 & 0.1024608 \\ 0.40 & 0.0955041 & 0.0955042 & 0.0957231 \\ 0.45 & 0.0891811 & 0.0891812 & 0.0893593 \\ 0.50 & 0.0831113 & 0.0831114 & 0.0832555 \\ 0.55 & 0.0772044 & 0.0772045 & 0.0773199 \\ 0.60 & 0.0713808 & 0.0713809 & 0.0714721 \\ 0.65 & 0.0655634 & 0.0655634 & 0.0656341 \\ 0.70 & 0.0596690 & 0.0596691 & 0.0597223 \\ 0.75 & 0.0535974 & 0.0535974 & 0.0536360 \\ 0.80 & 0.0472117 & 0.0472117 & 0.0472381 \\ 0.85 & 0.0402973 & 0.0402973 & 0.0403136 \\ 0.90 & 0.0324507 & 0.0324507 & 0.0324593 \\ 0.95 & 0.0226451 & 0.0226451 & 0.0226480 \end{array} \right)$$

So, it seems that we have a lower and an upper bound.

$$\color{blue}{ x^{\frac{1}{4 n}}\sqrt{\frac{-\log (x)}{n}} < \sqrt{1-x^{\frac{1}{n}}} <\sqrt{\frac{-\log (x)}{n}} }$$

Edit

What Barry Cipra proposed seems to be significantly better.

Answer 2

For some purposes, the bound $\sqrt{1-x^{1/n}}\le1$ is good enough. Better, of course, is

$$\sqrt{1-x^{1/n}}\le\sqrt{1-x}$$

But as the OP points out (in comments to the original version of this answer), neither of these bounds is very good when $n$ is large, except at $x=0$ (in both cases) and near $x=1$ (in the latter case).

A better bound -- and an improvement on Claude Leibovici's $\sqrt{|\log x|\over n}$ result -- can be obtained from the inequality

$$1-e^{-z}\le{2z\over2+z}\quad\text{for }z\ge0$$

with $z=-(\log x)/n$. We obtain

$$\sqrt{1-x^{1/n}}\le\sqrt{2|\log x|\over2n+|\log x|}$$

The main advantage of this bound occurs for $x\approx0$: We have ${2|\log x|\over2n+|\log x|}\le2$ for all $x\gt0$, whereas for any fixed $n$, we have ${|\log x|\over n}\to\infty$ as $x\to0^+$.

Answer 3

I prefer to add another answer to cover what seems to be the real problem.

In comments, @dohmatob explained that the real problem was for $$\sqrt{1-e^{-\frac{\log \left(\frac{1}{x}\right)+\sqrt{\log \left(\frac{1}{x}\right)}}{2 n}}}$$ which I shall consider in the form $$\sqrt{1-e^{-\frac{k}{ n}}} \qquad \text{with}\qquad k=\frac 12 \left(\log \left(\frac{1}{x}\right)+\sqrt{\log \left(\frac{1}{x}\right)} \right)$$

For sure, we could adapt Barry Cipra' solution for the initial problem.

Using the simplest Padé approximant, we can write $$f_n=\sqrt{1-e^{-\frac{k}{ n}}}\sim \frac{4 \sqrt{k n}}{k+4 n}=g_n$$ and all calculations seem to confirm that $f_n < g_n$. $$\left( \begin{array}{ccccccc} x & f_{10} & g_{10} & f_{20} & g_{20} & f_{30} & g_{30} \\ 0.01 & 0.53524626 & 0.53578216 & 0.39408621 & 0.39419365 & 0.32622013 & 0.32626080 \\ 0.02 & 0.50506690 & 0.50546031 & 0.37002676 & 0.37010438 & 0.30577897 & 0.30580819 \\ 0.03 & 0.48561695 & 0.48593657 & 0.35472284 & 0.35478530 & 0.29283270 & 0.29285614 \\ 0.04 & 0.47085583 & 0.47112748 & 0.34320523 & 0.34325796 & 0.28311638 & 0.28313612 \\ 0.05 & 0.45877877 & 0.45901579 & 0.33384049 & 0.33388625 & 0.27523249 & 0.27524959 \\ 0.10 & 0.41697472 & 0.41711869 & 0.30179781 & 0.30182514 & 0.24835956 & 0.24836972 \\ 0.15 & 0.38861766 & 0.38871761 & 0.28035851 & 0.28037729 & 0.23046082 & 0.23046778 \\ 0.20 & 0.36609954 & 0.36617301 & 0.26348492 & 0.26349862 & 0.21641539 & 0.21642045 \\ 0.25 & 0.34685670 & 0.34691237 & 0.24916233 & 0.24917265 & 0.20451995 & 0.20452376 \\ 0.30 & 0.32967964 & 0.32972255 & 0.23644690 & 0.23645482 & 0.19397844 & 0.19398135 \\ 0.35 & 0.31388484 & 0.31391824 & 0.22480867 & 0.22481481 & 0.18434477 & 0.18434702 \\ 0.40 & 0.29903566 & 0.29906174 & 0.21391134 & 0.21391612 & 0.17533648 & 0.17533823 \\ 0.45 & 0.28482423 & 0.28484459 & 0.20351963 & 0.20352335 & 0.16675646 & 0.16675782 \\ 0.50 & 0.27101288 & 0.27102869 & 0.19345356 & 0.19345643 & 0.15845436 & 0.15845542 \\ 0.55 & 0.25740031 & 0.25741248 & 0.18356236 & 0.18356457 & 0.15030471 & 0.15030552 \\ 0.60 & 0.24379833 & 0.24380758 & 0.17370679 & 0.17370847 & 0.14219205 & 0.14219266 \\ 0.65 & 0.23001149 & 0.23001837 & 0.16374396 & 0.16374521 & 0.13399839 & 0.13399884 \\ 0.70 & 0.21581323 & 0.21581822 & 0.15351006 & 0.15351096 & 0.12558895 & 0.12558927 \\ 0.75 & 0.20091002 & 0.20091350 & 0.14279461 & 0.14279523 & 0.11679107 & 0.11679129 \\ 0.80 & 0.18487420 & 0.18487649 & 0.13129283 & 0.13129324 & 0.10735521 & 0.10735536 \\ 0.85 & 0.16699267 & 0.16699404 & 0.11849837 & 0.11849861 & 0.09686737 & 0.09686746 \\ 0.90 & 0.14583635 & 0.14583704 & 0.10339861 & 0.10339873 & 0.08450015 & 0.08450020 \\ 0.95 & 0.11744221 & 0.11744245 & 0.08318823 & 0.08318827 & 0.06796219 & 0.06796220 \\ 0.96 & 0.10986307 & 0.10986324 & 0.07780275 & 0.07780278 & 0.06355780 & 0.06355781 \\ 0.97 & 0.10097970 & 0.10097981 & 0.07149485 & 0.07149487 & 0.05840022 & 0.05840022 \\ 0.98 & 0.08991159 & 0.08991165 & 0.06364157 & 0.06364158 & 0.05198069 & 0.05198069 \\ 0.99 & 0.07416134 & 0.07416137 & 0.05247613 & 0.05247613 & 0.04285642 & 0.04285642 \end{array} \right)$$

In terms of approximation, we can still improve. Below are given some expressions $$\sqrt{\frac{k}{n}}\, \frac{24 n-k}{24 n+5k}$$ $$\sqrt{\frac{k}{n}}\, \frac{480 n^2-48 k n+7k^2}{480 n^2+72kn}$$