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**Is "Zeta regularization" real??**

in many pages of the web i have found the intringuing result

[tex] \sum _{n=0}^{\infty} n^{s}= \zeta (-s) [/tex]

but the first series on the left is divergent ¡¡¡ for s >0 at least

other webpages use even more weird results

[tex] \sum _{n=0}^{\infty} h^{s+1}(a/h + n)^{s}\approx \int_{0}^{\infty}dx (a+x)^{s} [/tex] (h is an step)

but can we rely on these results for divergent series and integrals ? ,if so why there are still unsolved divergencies.