Does the series $\sum_{n=1}^\infty a_n$ converge? $ a_1=1,\ a_2=\sin(a_1),\dots, a_n =\sin(a_{n-1}) $

Asked Feb 03 '20 at 09:55

Active Feb 03 '20 at 11:08

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Recently, I came across an interesting problem. I haven't yet managed to solve it, but I really want to find out the solution.

Does the series $\sum_{n=1}^\infty a_n$ converge? $$ a_1=1,\ a_2=\sin(a_1),\dots, a_n =\sin(a_{n-1}) $$

I tried to use inequality $x\geqslant \sin(x)$ and prove by induction that $a_k\geqslant a_{k+1}$, but I didn't succeed. Anyway, I think that the sequence $\left\{a_k\right\}_{k=1}^\infty$ is monotonically decreasing, and it might be important.
As for the ratio test and the root test, I couldn't apply them here.

Could someone help me?

edited Feb 03 '20 at 11:08

jimjim

9,675

asked Feb 03 '20 at 09:55

Bonrey

1,294

2

$a_k>sin(a_k)=a_{k+1}\ \Rightarrow \ a_k>a_{k+1}$
obviously when $k\rightarrow\ \infty$ , then $a_k=sin(a_k)$ , therefore $a_k $tends to zero.But this all does not mean the series converges.
– aryan bansal Feb 03 '20 at 10:15
It's necessary that the sequence ${a_k}$ converges to 0, and it does. Ration test should work to show that the series converges. – Mathy Feb 03 '20 at 10:17
6

Show that $a_n\to0$, then show that $\lim\limits_{x\to0}\left(\frac1{\sin^2(x)}-\frac1{x^2}\right)=\frac13$. Apply Stolz-Cesàro to show that $a_n\sim\sqrt{\frac3n}$. – robjohn Feb 03 '20 at 10:53
1

Does this answer your question? Determine if this series converges or diverges – found with Approach0 – Martin R Feb 03 '20 at 12:10
@MartinR: Thanks for that link. I will post my answer there. – robjohn Feb 03 '20 at 12:16
Thank you, everyone! – Bonrey Feb 03 '20 at 16:46

Does the series $\sum_{n=1}^\infty a_n$ converge? $ a_1=1,\ a_2=\sin(a_1),\dots, a_n =\sin(a_{n-1}) $

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