Evaluate the sum

Calculus Level 5

Let \[S=\sum_{n=1}^\infty (-1)^{n-1} \ln{\left(\frac{n+1}{n}\right)}\]

Find the closed form for \(S\), and enter your answer as \(\lfloor 10^6\times S \rfloor\).

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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