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\begin{align*} \sum_{k=1}^{500}\frac{1}{2k-1} - \sum_{k=2}^{501}\frac{1}{2k-1} &= \frac{1}{2(1)-1} +\sum_{k=2}^{500}\frac{1}{2k-1} - \sum_{k=2}^{501}\frac{1}{2k-1}\\ &= \frac{1}{1} + \sum_{k=2}^{500}\frac{1}{2k-1} - \left(\sum_{k=2}^{500}\frac{1}{2k-1} + \frac{1}{2(501) - 1}\right)\\ &= \frac{1}{1} + \sum_{k=2}^{500}\frac{1}{2k-1} - \sum_{k=2}^{500}\frac{1}{2k-1}- \frac{1}{1001}\\ &= \frac{1}{1} - \frac{1}{1001} \end{align*}
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Fri, 02 Mar 2018 22:07 GMT