Why can't an argument for the Riemann Zeta function be 1? What happens if we take Re(s)=1?

Question

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?

Answer 1

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.