Does it diverge or converge?

Calculus Level 3

Does the following series diverge or converge?

\[\sum _{n=1}^{\infty } \dfrac{\sin(\text{ln}(n))}{n}.\]


The Basics:
Let \(\sum_{n=m}^\infty a_n\) be a formal infinite series. For any integer \(N\geqslant m\), we define the \(N^{\text{th}}\) partial sum \(S_N\) of this series to be \(S_N:=\sum_{n=m}^N a_n\); of course, \(S_N\) is a real number. If the sequence \((S_N)_{n=m}^\infty\) converges to some limit \(L\) as \(N\to\infty\), then we say that the infinite series \(\sum_{n=m}^\infty a_n\) is convergent, and converges to \(L\); we also write \(L=\sum_{n=m}^\infty a_n\), and say that \(L\) is the sum of the infinite series \(\sum_{n=m}^\infty a_n\). If the partial sums \(S_N\) diverge, then we say that the infinite series \(\sum_{n=m}^\infty a_n\) is divergent.

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