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I do not understand a small step in a proof I'm reading at the moment. Why are the following things equal?

$$\sum_{k=1}^{n} \frac{1}{2k-1} - \frac{1}{2} \sum_{k=1}^{n} \frac{1}{k} = \sum_{k=1}^{2n} \frac{1}{k} - \sum_{k=1}^{n} \frac{1}{k}$$

rehband
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    $\displaystyle\sum_1^n\frac{1}{2k-1}-\frac12\sum_1^n\frac1k=\sum_1^n\frac{1}{2k-1}-\frac1{2k}=\sum_1^n\frac{1}{2k-1}+\frac1{2k}-\frac{2}{2k}=\sum_1^{2n}\frac1k- \sum_{1}^{n}\frac1k$ – OBDA Jul 22 '14 at 15:38

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Separating out the odd & the even terms in the denominator, $$\sum_{k=1}^{2n}\frac1k=\sum_{k=1}^n\left(\frac1{2k-1}+\sum_{k=1}^n\frac1{2k}\right)$$

$$=\sum_{k=1}^n\frac1{2k-1}+\sum_{k=1}^n\frac1{2k}$$

$$=\sum_{k=1}^n\frac1{2k-1}+\frac12\sum_{k=1}^n\frac1k$$

lab bhattacharjee
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