proof explanation - Value of the Riemann Zeta Function for

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

Riemann Zeta Function - an overview

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PDF] Separation of zeros of the Riemann zeta-function

On the approximation of Riemann's Zeta Function for S=3 (Apéry's constant)

The Riemann Hypothesis Resolved. I will explain what the Riemann…, by Fayez A. Alhargan, PhD

The holy grail of mathematics: animated visualization of the Riemann Zeta zeros

3Blue1Brown - Visualizing the Riemann zeta function and analytic continuation

Solved 10. Prove that Riemann zeta function ζ(x)=∑n=1+∞nx1

New proof that the sum of natural numbers is -1/12 of the zeta function