Playing with desmos I have accidentally noticed that the sequence of partial sums $$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$

is bounded.

However, I did not succeed in proving this statement.

My main idea was to show that there exists some constant $C > 0$ such that

$$\forall N \geq 1 \; \forall k \geq 0 \; : \left\vert \sum_{n=1}^{N} (\sin(n))^{2k+1}\right\vert \leq C$$

And then using the Taylor series

$$\sin(\sin(n)) = \sum_{m=0}^{\infty}\frac{(-1)^m}{(2m+1)!}(\sin(n))^{2m+1}$$

bound the partial sums defined above by $Ce$.

Could you please help me or suggest the techniques which would be useful in proofs of such statements.

Playing with desmos I have accidentally noticed that the sequence of partial sums $$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$

is bounded.

However, I did not succeed in proving this statement.

My main idea was to show that there exists some constant $C > 0$ such that

$$\forall N \geq 1 \; \forall k \geq 0 \; : \left\vert \sum_{n=1}^{N} (\sin(n))^{2k+1}\right\vert \leq C$$

And then using the Taylor series

$$\sin(\sin(n)) = \sum_{m=0}^{\infty}\frac{(-1)^m}{(2m+1)!}(\sin(n))^{2m+1}$$

bound the partial sums defined above by $Ce$.

Could you please help me or suggest the techniques which may be useful in proving statements of this kind.

Playing with desmos I have accidentally noticed that the sequence of partial sums $$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$

is bounded.

However, I did not succeed in the proof of this statement.

My main idea was to show that there exists some constant $C > 0$ such that

$$\forall N \geq 1 \; \forall k \geq 0 \; : \left\vert \sum_{n=1}^{N} (\sin(n))^{2k+1}\right\vert \leq C$$

And then using the Taylor series

$$\sin(\sin(n)) = \sum_{m=0}^{\infty}\frac{(-1)^m}{(2m+1)!}(\sin(n))^{2m+1}$$

bound the partial sums defined above by $e^C$.

Could you please help me or suggest the techniques which would be useful in proofs of such statements.

Playing with desmos I have accidentally noticed that the sequence of partial sums $$\left\{ \sum_{n=1}^{N}\sin(\sin(n)) : N\geq 1 \right\}$$

is bounded.

However, I did not succeed in proving this statement.

My main idea was to show that there exists some constant $C > 0$ such that

$$\forall N \geq 1 \; \forall k \geq 0 \; : \left\vert \sum_{n=1}^{N} (\sin(n))^{2k+1}\right\vert \leq C$$

And then using the Taylor series

$$\sin(\sin(n)) = \sum_{m=0}^{\infty}\frac{(-1)^m}{(2m+1)!}(\sin(n))^{2m+1}$$

bound the partial sums defined above by $Ce$.

Could you please help me or suggest the techniques which would be useful in proofs of such statements.