Does $\sum a_n$ converge if $a_n = \sin( \sin (...( \sin(x))...)$

• Classicaly, $\displaystyle a_n=\sqrt\frac{3}{n}+o\left(\sqrt\frac{3}{n}\right)$

The series $\sum a_n$ is therefore divergent.

• Using the previous asympotical estimate, $\sum a_n^r$ converges iff $1/2-r> 1$

Proof of the first claim:

Notation: Given two sequences $u_n$ and $v_n$, we note $u_n\sim v_n$ iff $ \lim_{n\to \infty}\frac{u_n}{v_n}=1$

Now, back to the asymptotics:

$$\begin{align} \frac{1}{a_{n+1}^2}-\frac{1}{a_n^2}=\frac{1}{\sin(a_n)^2}-\frac{1}{a_n^2}&=\frac{1}{a_n^2-\frac{a_n^4}{3}+o(a_n^4)}-\frac{1}{a_n^2}\\&=\frac{1}{a_n^2}\left(\frac{1}{1-\frac{a_n^2}{3}+o(a_n^2)}-1\right)\\&=\frac13+o(1) \\\end{align}$$

By Cesaro mean theorem, $\displaystyle \frac{1}{n}\sum_{k=1}^n \left(\frac{1}{a_{k+1}^2}-\frac{1}{a_k^2}\right)=\frac13+o(1)$

Hence $\displaystyle a_n^2=\frac{1}{\frac{n}{3}+o(n)}=\frac{3}{n}\left(1+o(1)\right)$

Hence $\displaystyle a_n=\sqrt\frac{3}{n}+o\left(\sqrt\frac{3}{n}\right)$

The estimate can be refined as $$a_n=\sqrt\frac{3}{n}-\frac{3\sqrt{3}}{10}\frac{\log(n)}{n\sqrt{n}}+o\left(\frac{\log(n)}{n\sqrt{n}}\right)$$

Proof of the estimate on partial sums

Lemma: Let $u_n$ and $v_n$ be sequence such that $u_n\sim v_n$, and $\forall n\in \mathbb N,v_n\geq 0$. If $\sum_{n\geq 1}v_n$ diverges, so does $\sum_{n\geq 1}v_n$ and $\sum_{k=1}^nu_k \sim \sum_{k=1}^nu_k$

We've proven so far that $\displaystyle a_n\sim \sqrt\frac{3}{n}$.

Hence, by the lemma, $\displaystyle S_n \sim \sum_{k=1}^n \sqrt\frac{3}{k}$

To estimate the last sum, an integral method comes in handy: indeed

$$\forall n\geq 2, \int_n^{n+1}\frac{1}{\sqrt{t}}dt \leq \frac{1}{\sqrt{n}} \leq \int_{n-1}^n \frac{1}{\sqrt{t}}dt$$

Summing these inequalities yields $$\sum_{k=1}^n \sqrt\frac{3}{k} \sim 2\sqrt{3n} $$

Hence $S_n \sim 2\sqrt{3n}$