\[\displaystyle{ S }_{ N }=\sum _{k=1}^{N}\frac1k \]

Let \(S_N\) satisfy the equation above. What is the value of \(\displaystyle \lim _{ n\rightarrow \infty }{ \left( { S }_{ 2n }-{ S }_{ n } \right) }\)?

Give your answer to three decimal places.

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