# How maths can help to compute convergence?

$r'(\theta)^2 + r(\theta)^2 = \theta^2,\quad r(t=0)=0\tag{1}$

There is an interesting approach to prove that the solutions of the equation $(1)$ have power series representations of form $r(\theta) = \pm \ \left(\frac{\theta^2}{2}-\frac{\theta^4}{32}+\frac{\theta^6}{768}+\cdots\right)$, and the power series convergence. (More details

However, I am still unable to compute the radius of convergence analytically. Any areas of math to solve the problem? Any help, thoughts are highly welcomed.

• Doesn't @RobertBryant give all the relevant references in his very nice answer to the original question? (mathoverflow.net/questions/121402/…)
– Igor Rivin
Aug 7 '13 at 17:44
• @Igor Revin I understand that this is a nice prove, but what about the radius?
– Mikhail Gaichenkov
Aug 7 '13 at 18:03
• I don't know who this Igor Revin is, but as for @RobertBryant's proof, if you read it to the end, he gives a reference addressing the very question you asks.
– Igor Rivin
Aug 8 '13 at 2:02
• @Igor Rivin. The reference just explains the power series convegence, but I cannot see that the radius is, I guess, 7/2?
– Mikhail Gaichenkov
Aug 8 '13 at 4:15
• Well, the kind of mathematics can prove the convergence, but I cannot see any way to compute the radius of convergence. For eg how to prove it is 7/2? Aug 8 '13 at 18:06

The reason you didn't get a satisfactory answer so far is that your problem is not a bona fide initial problem and cannot be turned into one using suitable substitutions and the like. When you introduce, e.g., a new independent variable $u$ by means of $t:=\sqrt{2u}$ you get an "initial value problem" of the form $$\dot r(u)=\sqrt{1-{r^2(u)\over u}},\quad r(0)=0\ .$$ Changing the initial condition to $r(0)=10^{-100}$ lets the solution disappear, contrasting the case of "ordinary" IVP's, where the solutions depend continuously on the initial conditions.
Nevertheless, the solution you are interested in can be written as $$r(t)=\sum_{k=1}^\infty a_k t^{2k}\ ,$$ whereby the $a_k$ satisfy the following recursion: $$a_1={1\over2},\quad a_n=-{1\over 4n}\left(\sum_{k=1}^{n-1}a_k a_{n-k}+4\sum_{k=2}^{n-1}k(n+1-k)a_ka_{n+1-k}\right)\qquad(n\geq2)\ .$$ Using this recursion and putting $b_n:=|a_n|^{-1/2n}$ one obtains the following plot (note that $\lim\inf_{n\to\infty} b_n$ is the radius of convergence you are after): 