Does it diverge or converge?

Calculus Level 3

Does the following series diverge or converge?

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


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

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