=1-\frac{1}{2} + \frac{1}{3} + \frac{1}{4}\cdots -\frac{1}{2n-1}
We have RHS = \sum_{k=1}^{n}\frac{1}{2k-1} - \sum_{k=1}^{n}\frac{1}{2n}
=\sum_{k=1}^{n}\frac{1}{2n-1} + \sum_{k=1}^{n}\frac{1}{2k} - 2 \sum_{k=1}^{n-1}\frac{1}{2k}
=\sum_{k=1}^{2n-1}\frac{1}{k}- \sum_{k=1}^{n-1}\frac{1}{k}
=\sum_{k=n}^{2n-1}\frac{1}{k}=LHS
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