Convergence of $\prod_{n=1}^\infty n\sin\frac{x}{n}$

Question

If $\prod_{n=1}^\infty n\sin\frac{x}{n}$ converges to a nonzero value, then $x=1$.

The other direction is done using Taylor series, but how do I prove this?

Answer 1

If $\prod_{n=1}^\infty a_n$ converges to a nonzero value, then $\lim_{n\to \infty} a_n = 1$.

(The proof of this is similar to the standard proof that if $\sum_{n=1}^\infty a_n$ converges, then $\lim_{n\to \infty} a_n = 0$. In this case, if $P_n$ is the $n$th partial product, and the product converges to $L \ne 0$, then $a_n = \frac{P_n}{P_{n-1}} \to \frac{L}{L} = 1$ as $n \to \infty$.)