[hkjjutrv] Riemann zeta convergence

Using tricks, namely analytic continuation, we can get the natural numbers 1 + 2 + 3 + ... to sum to a mindboggling finite value, namely zeta(-1) = -1/12.  However, no tricks can get the reciprocals of the natural numbers, the harmonic series, 1/1 + 1/2 + 1/3 + ... = zeta(1) to converge.  It is truly a pole.  This is surprising because the harmonic series seems less outrageously infinite than the sum of the natural numbers.

What characterizes divergent series which can be summed versus those which cannot?

Update: some tricks can make the harmonic series to sum to gamma, the Euler-Mascheroni constant.