Like origami art, where we can fold a paper to create a boat and then refold the same paper differently to build a totally different thing (for example, a bird), the simple steps below show that we can unfold an infinite series (commonly known as Riemann's zeta function) and then refold it to get a finite series. It is fascinating to see unfolding a divergence function in 6 steps and then refolding the same function in 6 steps gives us a convergent function. This is the most elegant method to analytically continue Riemann's zeta function to the critical strip because it shows mathematics overlaps with art.

Note: Consider the infinite series transition below for step 6. (s=1 has been shown for simplicity) :

https://en.wikipedia.org/wiki/Dirichlet_eta_function

https://mathworld.wolfram.com/DirichletEtaFunction.html

https://en.wikipedia.org/wiki/Telescoping_series

Server IP: 50.18.238.17