Infinity minus infinity

Calculus Level 2

SN=k=1N1k\displaystyle{ S }_{ N }=\sum _{k=1}^{N}\frac1k

Let SNS_N satisfy the equation above. What is the value of limn(S2nSn)\displaystyle \lim _{ n\rightarrow \infty }{ \left( { S }_{ 2n }-{ S }_{ n } \right) }?

Give your answer to three decimal places.

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