The figure below, made of right triangles with sides $1,\sqrt{n}$ and $\sqrt{n+1}$ spiraling around a common vertex (opposite side 1) beginning at $n=1$ is the (classical) "Theodorus spiral" ; see for example https://mathematica.stackexchange.com/q/66969 .
I would like to establish that the limit curve is an Archimedean spiral.
Let us recall that such is spiral has a polar equation of the form : $$\dfrac{\theta}{r}=a \ \ \text{ for some constant} \ a \ \ \ \ \ \ \ (1)$$
$\bf{My} \ \bf{attempts}$ : My idea is to bring back this issue to the fact that the following limit exists :
$$a=\lim_{N \to \infty} \dfrac{1}{\sqrt{N}} \sum_{n=1}^N \mathrm{atan} \left(\dfrac{1}{\sqrt{n}} \right) \ \ \ \ (2)$$
where the summation gives the general polar angle $\theta$ and $\sqrt{N}$ corresponds to radius $r$ in (1). Numerical computations give evidence for that with a value of $a$ looking slightly below $2$.
I have taken two ways :
- I have transformed the summation in (2) using relationship $$\mathrm{atan}(a)+\mathrm{atan}(b)=\mathrm{atan}\dfrac{a+b}{1-ab} \ \ \ \ \ (3)$$ in a recurrent way, giving a single expression $\mathrm{atan}\dfrac{u_N}{v_N}$ through the double sequence :
$$\begin{cases}u_{n}&=&\sqrt{n} \ u_{n-1} & + & v_{n-1}\\ v_{n}&=&-u_{n-1}&+&\sqrt{n} \ v_{n-1} \end{cases}, \ \ u_1=v_1=1 \ \ \ \ (4)$$
but I am blocked there...
- I have attempted a different way by using Euler-Maclaurin summation formula (see Appendix below) using the fact that an antiderivative of function defined by $f(x)=\mathrm{atan}(\tfrac{1}{\sqrt{x}})$ is given by $$F(x)=\sqrt{x}+x \ \mathrm{atan}(\tfrac{1}{\sqrt{x}})-\mathrm{atan}(\sqrt{x}).$$
together with the development:
$$\mathrm{atan}(\tfrac{1}{\sqrt{x}})=\tfrac{1}{\sqrt{x}}-\tfrac{1}{3 x \sqrt{x}}+\tfrac{1}{5x^2\sqrt{x}}- ...$$
which is a more rewarding direction...
Maybe, there other ways to prove that we have asymptotically an Archimedean spiral...
I would be grateful for any hint...
In fact, I discovered using the "Theodorus" keyword (that I hadn't when writing a first draft of this question) that the article https://en.wikipedia.org/wiki/Spiral_of_Theodorus provides a lot of information about this (not so trivial) issue...
Edit (Dec. 22, 2022) : I just discovered the exceptional book "Spirals from Theodorus to Chaos" (Philip J. Davis with contributions from Walter Gautschi, A. Iserles AK Peters, 1993) with answers to my questioning... and much more, all in a pleasant conversational style. I strongly advise it.
Appendix on Euler-Maclaurin formula : for any integers $a$ and $b$ with $0<a<b$ :
\begin{eqnarray} \sum_{k = a}^{b} f(k) = \int_{a}^{b} f(t) \, dt + \tfrac12 (f(a) + f(b)) + \sum_{n = 1}^{N} \frac{B_{2n}}{(2n)!} ( f^{(2n-1)}(b) - f^{(2n-1)}(a) ) + R_{N}, \end{eqnarray} where $B_{n}$ is the $n$th Bernoulli number with $B_{2} = \tfrac{1}{6}$, $B_{4} = -\tfrac{1}{30}$, $B_{6} = \tfrac{1}{42}$, etc... and \begin{align} |R_{N}| \leq \frac{|B_{2N} |}{(2n)!} \int_{a}^{b} | f^{(2N)}(t) | \, dt. \end{align} for any arbitrary positive integer $N$.
