\[ { n \choose 1} - \left (1 + \frac 1 2 \right )
{ n \choose 2} + \left (1 + \frac 1 2 + \frac 1 3 \right )
{ n \choose 3} - \ldots \\ + (-1)^{n-1} \left (1 + \frac 1 2 + \ldots + \frac 1 n \right )

{ n \choose n} \]

If the value of the expression above equals to \(S\), then what can we conclude?

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