# 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*.