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Question Number 32274 by abdo imad last updated on 22/Mar/18
prove that  Σ_(k=0) ^n    (2^k /(x^2^k   +1)) = (1/(x−1)) − (2^(n+1) /(x^(2^(n+1)  ) −1))  .
$${prove}\:{that}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\:\frac{\mathrm{2}^{{k}} }{{x}^{\mathrm{2}^{{k}} } \:+\mathrm{1}}\:=\:\frac{\mathrm{1}}{{x}−\mathrm{1}}\:−\:\frac{\mathrm{2}^{{n}+\mathrm{1}} }{{x}^{\mathrm{2}^{{n}+\mathrm{1}} \:} −\mathrm{1}}\:\:. \\ $$

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