Draw a circle in the first quadrant tangent to $y=1/x, y=0$ and $x=0$, then continue drawing circles that are tangent to the previous circle, $y= 1/x$ and $y=0$.

What is the radius of the $n$-th circle?

Radius of first circle is $2 - \sqrt{2}$.

but even calculating second radius is difficult. It is over a month I am thinking on it.

I think question title is obvious. assume we have a rectangular hyperbola chart.and we draw largest circle which fits under $y=1/x, y=0$ and $x=0$. then we continue drawing circles which are tangent to previous circle, $y= 1/x$ and $y=0$. Question is: what is radius of $n$-th circle.

Radius of first circle is $2 - \sqrt{2}$.

but even calculating second radius is impossible. It is over a month I am thinking on it.

PS 1: I try to find the line which connect all of circle's centers. if we call if f(x), it is clear that $$\lim_{x\to \infty}\frac{f(x)}{(1/x)}=\frac{1}{2}$$ another thing is following formula. the centers of circles that are in equal distance from y = 0 and first circle. (second circle center in on this line:) $$y=\frac{\left(r_{0}-x\right)^{2}}{4r_{0}}$$$$r_{0}=2-\sqrt{2}$$ and I get 6 formulas as follows: there are three points which are hit points of three curves. first circle and y=1/x hit point is (1,1) but two other points should be found. each of points satisfy curve formulas. also in eqach derivative of both curves are equal. and also, distance of two first circle's centers is r0+r1 which r1=y1.

I think question title is obvious. assume we have a rectangular hyperbola chart.and we draw largest circle which fits under $y=1/x, y=0$ and $x=0$. then we continue drawing circles which are tangent to previous circle, $y= 1/x$ and $y=0$. Question is: what is radius of $n$-th circle.

Radius of first circle is $2 - \sqrt{2}$.

but even calculating second radius is impossible. It is over a month I am thinking on it.

Draw a circle in the first quadrant tangent to $y=1/x, y=0$ and $x=0$, then continue drawing circles that are tangent to the previous circle, $y= 1/x$ and $y=0$.

What is the radius of the $n$-th circle?

Radius of first circle is $2 - \sqrt{2}$.

but even calculating second radius is difficult. It is over a month I am thinking on it.

I think question title is obvious. assume we have a rectangular hyperbola chart.and we draw largest circle which fits under y=1/x, y=0 and x=0. then we continue drawing circles which are tangent to previous circle, y= 1/x and y=0. question is  : what is radius of Nth circle.

link to question : link

radius of first circle is $2 - \sqrt{2}$.

but even calculating second radius is impossible. it is over a month I am thinking on it.

I think question title is obvious. assume we have a rectangular hyperbola chart.and we draw largest circle which fits under $y=1/x, y=0$ and $x=0$. then we continue drawing circles which are tangent to previous circle, $y= 1/x$ and $y=0$. Question is: what is radius of $n$-th circle.

Radius of first circle is $2 - \sqrt{2}$.

but even calculating second radius is impossible. It is over a month I am thinking on it.