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Some arithmetic Identities
10-09-2010, 06:30 PM
Post: #1
Some arithmetic Identities
(1) $\displaystyle{\sum_{k=1}^{2n}\frac{(-1)^{k-1}}{k}=\sum_{k=1}^n \frac{1}{n+k}}$
$\quad\quad\displaystyle{1-\frac{1}{2}+\frac{1}{3}-\cdots+\frac{1}{2n-1}-\frac{1}{2n} = \frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+k}+\cdots+\frac{1}{2n}}$

(2) $\displaystyle{\sum_{k=2}^{2n}\frac{(-1)^k (2n+1-k)}{k}=\sum_{k=1}^n\frac{2k-1}{n+k}}$
$\quad\quad\displaystyle{\frac{2n+1-2}{2}-\frac{2n+1-3}{3}+\cdots-\frac{2}{2n-1}+\frac{1}{2n}=\frac{1}{n+1}+\cdots+\frac{2k-1}{n+k}+\cdots+\frac{2n-1}{2n}}$
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Some arithmetic Identities - elim - 10-09-2010 06:30 PM
RE: Identities (1) and (2) - elim - 10-13-2010, 06:22 PM

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