If $s=1$, then the series equals to $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...\to \infty$
This certainly does seem to be a convergent series. Why doesn't it have a limit?
2026-04-06 03:40:00.1775446800
Why can't an argument for the Riemann Zeta function be 1? What happens if we take Re(s)=1?
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The series is the harmonic series, and it's pretty easy to see that it diverges.
One way to see this is to notice that the series is bounded by
$$ \int_1^\infty \frac{1}{x} = ln(x) \bigg|_1^\infty = \infty $$
Since the integral is divergent, the sum is also divergent.