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Tuesday, June 28, 2016

2016/058) Show that \frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\cdots\frac{1}{2n-1}

=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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