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 hypothesis - Wikipedia

A Centuries-Old Mathematical Puzzle May Finally Have an Answer - Science in the News

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

Particular values of the Riemann zeta function - Wikipedia

Riemann zeta function - Wikipedia

Riemann hypothesis - Wikipedia

The Riemann Zeta Function and Probability Distributions – Almost Sure

A Proof of the Riemann Hypothesis

Doc Madhattan: Riemann hypothesis

Prime number mystery divides math world

Particular values of the Riemann zeta function - Wikipedia