If \( s_{p} \) denotes the sum of the series \( 1+r^{p}+r^{2 p}+\ldots \) to \( \infty \) and \(...

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If \( s_{p} \) denotes the sum of the series \( 1+r^{p}+r^{2 p}+\ldots \) to \( \infty \) and \( s_{p} \) the sum of the series \( 1-r^{p}+r^{2 p}-\ldots \) to \( \infty \), prove that \( S_{p}+s_{p}=2 S_{2 p} \).
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