... the solution of which is $\displaystyle y = \frac{c}{\sqrt{x}}$, so that we can suppose $\displaystyle a_{n} \sim r_{n}= \frac{c}{\sqrt{n}}$. If we suppose that $r_{500000}= \frac{1}{1000}$ then is $\displaystyle c = \frac{1}{\sqrt{2}}$. In the following table the first values os $a_{n}$ and $r_{n}$ are reported...
$a_{1}= 1,\ r_{1} = .70710678...$
$a_{2}= .5,\ r_{2} = .5$
$a_{3}= .375,\ r_{3} = .40824829...$
$a_{4}= .3125,\ r_{4} = .35355339...$
$a_{5}= .273438...,\ r_{5} = .31627766...$
$a_{6}= .246094...,\ r_{6} = .2886751...$
$a_{7}= .225586...,\ r_{7} = .2672612...$
$a_{8}= .209473...,\ r_{8} = .25$
$a_{9}= .196381...,\ r_{9} = .2357022...$
$a_{10}= .185471...,\ r_{10} = .2236067...$
It is clear from the table that for n 'large enough' the relative increments of the $a_{n}$ and $r_{n}$ are pratically the same and that is verified considering that is...