proof explanation - Value of the Riemann Zeta Function for negative integers - Mathematics Stack Exchange

I am reading the book "Riemann's Zeta Function" by H. M. Edwards. I had a confusion in Section 1.5, Page 12 at the derivation of the formula of Riemann Zeta Function at negative integers

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PDF) Computation of the Riemann zeta function in a complex domain

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How is the Riemann zeta function zero for negative even integers

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