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The sum $\sum\limits_{i=1}^{2013} \frac{\left(-1\right)^{i-1}}{i} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots - \frac{1}{2012} + \frac{1}{2013}$ equals $\text{\_\_\_\_\_\_\_\_\_\_\_\_}.$

$$ SOURCE: NSEJS 2013

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