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The following proof is fake, but I have no idea why it's not right (I know it's paradoxical, since $\zeta (1)$ diverges): $$\begin{align*}\zeta (1)&=\dfrac{2\zeta (1)}{2}\\&=\dfrac{\displaystyle\lim_{x\to 0}\zeta (1+x)+\displaystyle\lim_{x\to 0}\zeta (1-x)}{2}\\&\overset{!}{=}\displaystyle\lim_{x\to 0}\dfrac{\zeta (1+x)+\zeta (1-x)}{2}\\&=\gamma\approx 0.577.\end{align*}$$

Tannery's theorem is of the form $a\implies b$. Here, I did it the other way around: $b\implies a$. So, that's not equivalent (but is close) to Tannery's theorem. Where is the mistake? I suspect it's the "$!$", but which theorem does it refer to?

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    $\begingroup$ I think the limit $\lim_{s \to 1} \zeta(s)$ does not exists. $\endgroup$ – Botond Aug 23 '19 at 12:29
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    $\begingroup$ It's not right because the limits you refer to don't exist, so you can't use the usual theorems on them $\endgroup$ – Maxime Ramzi Aug 23 '19 at 12:29
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    $\begingroup$ To emphasize: IF $\lim_{n \to \infty} x_n = L$ AND $\lim_{n \to \infty} y_n = M$ THEN $\lim_{n \to \infty} (x_n + y_n) = L + M$. If you cannot verify the two hypotheses, then you are not justified in using the conclusion. $\endgroup$ – Lee Mosher Aug 23 '19 at 12:35
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    $\begingroup$ How do you define $\zeta (1)$ $\endgroup$ – reuns Aug 23 '19 at 12:39
  • $\begingroup$ The Cauchy' sprincipal value of $\zeta(1)$ is indeed $\gamma$. $\endgroup$ – Anixx Aug 24 '19 at 5:04
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You cannot use the rule $\lim\limits_{n \to \infty} (a_n + b_n) = \lim\limits_{n \to \infty} a_n + \lim\limits_{n \to \infty} b_n$ here because the limits do not exist. Like that you could create all kinds of wrong results ($\infty - \infty$ can be anything).

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    $\begingroup$ E.g., $\lim n+\lim(7-n)=7$. $\endgroup$ – Gerry Myerson Aug 23 '19 at 12:32

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