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

Special values of zeta functions – Bhāvanā

3Blue1Brown - Visualizing the Riemann zeta function and analytic continuation

Using Riemann Zeta Function and DoubleFloats to calculate Pi - Julia Community 🟣

On Cryptography and the Zeta Function, by Bob McElrath

Riemann Hypothesis: What Yitang Zhang's New Paper Means and Why You Should Care – The Wire Science

Lines in the complex plane where the Riemann zeta function ζ is real

Analytic Continuation and the Riemann Zeta Function – IB Maths Resources from Intermathematics

Riemann zeros from Floquet engineering a trapped-ion qubit

Riemann zeta function 1

The Riemann Hypothesis, explained

The Riemann Zeta Function

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

A New Approach To The Riemann Hypothesis Could Be Worth $1,000,000