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Jun 2 at 18:05 answer added Kamal Saleh timeline score: 0
May 27 at 20:06 comment added Kamal Saleh @StevenClark That's essentially the analytic continuation I used to come up with the identities in OP.
May 27 at 19:57 comment added Steven Clark Have you considered using the definition $$\sum _{n=1}^x \frac{(-1)^{n+1}}{n}=\log (2)-(-1)^x \Phi(-1,1,x+1)$$ where $\Phi$ is the Lerch Transcendent (see WolframAlpha)? There are several identities for the Lerch Transcendent on the Wolfram Function Site.
May 27 at 19:28 history edited Kamal Saleh
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May 27 at 19:22 history asked Kamal Saleh CC BY-SA 4.0