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From WolframAlpha it seems that $$ \frac{1}{2}=\sum_{k=1}^{\infty} \zeta(2k)-\zeta(2k+1) $$

Could someone provide a proof for this?

Thanks.

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  • $\begingroup$ Where in WA is written that? $\endgroup$ – DonAntonio Jan 13 '14 at 11:09
  • $\begingroup$ ...WA provided $\approx 0.4999999999999...$ not exactly $=1/2$. $\endgroup$ – Neves Jan 13 '14 at 11:13
  • $\begingroup$ Not that I asked, @Neves ... $\endgroup$ – DonAntonio Jan 13 '14 at 11:16
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    $\begingroup$ Are you sure that the summation starts at k=0 ? Not k=1 instead ? $\endgroup$ – Claude Leibovici Jan 13 '14 at 11:16
  • $\begingroup$ @claude, corrected that. $\endgroup$ – Neves Jan 13 '14 at 11:17
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Writing zeta-functions as series and changing the summation order does the trick.

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