Cauchy-sequence convergence monitor

What you are seeing: partial sums of a chosen series with the Cauchy width $w(N_0) = \max_{n,m \ge N_0} |a_n - a_m|$ as the convergence diagnostic. A sequence is convergent iff $w(N_0) \to 0$ as $N_0$ grows. Geometric and $\zeta(2)$ series shrink fast; the harmonic series fails to be Cauchy and its width grows logarithmically.