Cauchy Sequence Convergence Monitor

How can you tell a sequence converges without knowing what it converges to? Cauchy's answer: look at how tightly the terms bunch up far out. A sequence is Cauchy if, for every tolerance $\varepsilon$, there is a point $N_0$ beyond which every pair of terms is within $\varepsilon$ of each other, $|a_n - a_m| \lt \varepsilon$. The scene plots the sequence and a sliding tail window starting at $N_0$; the shaded strip is its width, the largest gap between any two terms past $N_0$. Slide $N_0$ out and watch the strip collapse toward zero, that is the Cauchy condition, and on the real line it is the same thing as converging. The harmonic sum is the cautionary case: its terms shrink, yet the tail width never falls below about $\ln 10$, so it is not Cauchy and does not converge.

Figure 1. Cauchy convergence monitor. Top: the sequence on a log index axis with the tail window from N₀ and its width (the largest gap between terms beyond N₀) against the ε tolerance. Bottom: the tail width versus N₀, falling to zero for a Cauchy sequence and flattening for the harmonic. Method: tail max minus min over a multiplicative window.

sequence

tail N₀10

ε3.2e-2

WHAT TO TRY

Slide the tail start N₀ outward: for a convergent sequence the width strip collapses toward zero, the terms bunch arbitrarily tightly.
Demand a smaller ε: you need a larger N₀ before the tail fits inside it, the response to the challenge that defines Cauchy.
Switch to the harmonic sum: the tail width never drops below about ln 10, so no N₀ ever satisfies a small ε. It is not Cauchy.
The Leibniz and zeta sequences converge slowly; their strips shrink, just more gradually than the geometric one.