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# On a nested sin sequence

Consider the sequence defined recursively as

Prove that .

Solution

Lemma: If is a sequence for which then

Proof: In Stolz theorem we set and .

It is easy to see that is is monotonically decreasing to zero. Moreover, an application of L’Hospital’s rule gives

Therefore

Now, due to the lemma we have and the result follows.

Remark : The asymptotic now follows to be .

Problem: Find what inequality should satisfy such that the series

converges.

## 1 Comment

1. We present a generalisation .

Claim: It holds that: