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--- pow:problem5s21 [2021/04/26 15:00] – created mazur
+++ pow:problem5s21 [2021/04/27 03:23] (current) – mazur
@@ Line 1: / Line 1: @@
+<box 80% round orange| Problem 5 (due Monday, April 26)>
+The sequence $(a_n)$ is defined recursively as follows: $a_1=1$, $a_{n+1}=\sin a_n$.
+Prove that the sequence $(\sqrt{n}a_n)$ converges and find its limit.
+</box>
+Two solutions were received, from John Mignone and Yuqiao Huang. Both are correct and are based on a
+result going back to Cauchy, which is a special case of Stolz theorem (solution by Yuqiao) and is
+related to Cesaro summation (John's solution). This is covered in our second solution. Our first solution
+is more direct (but longer) and shows that the sequence $\sqrt{n}a_n$ is increasing. For details
+see the following link {{:pow:2021sproblem5.pdf|Solution}}.