Sequences and convergence
The sequence $a_n = \dfrac{(-1)^n}{n}$ runs $-1, \frac{1}{2}, -\frac{1}{3}, \frac{1}{4}, \dots$ — it flips sign on every step, forever. Does it converge?
At $\varepsilon = 0.25$ the band around $L = 0$ traps the tail of $1/n$ from $n = 5$ onward — every later dot brightens and stays inside. Narrow the band to $\varepsilon = 0.05$ and the entry point retreats to $n = 21$, but it still exists. Now switch to $(-1)^n/n$: the dots hop across the limit line on every step, yet the stat reads the same $5$ and the same $21$. Switch to $1.1^n$ and the stat surrenders: never.
That is convergence. A sequence converges to $L$ when every epsilon band around $L$, however narrow you set it, eventually traps the sequence for good. The sign flips of $(-1)^n/n$ never mattered because the test only measures the distance $|a_n - L|$, and that distance sinks below any bound. What kills convergence is escaping the band, the way $1.1^n$ outruns every one. The formal definition says exactly what the instrument computes:
About this model
What it represents: the first $40$ terms of each sequence, an adjustable band $L \pm \varepsilon$, and a search for the first index after which every checked term stays strictly inside the band.
What it leaves out: infinity. The plot shows $40$ dots and the stat checks the first $100{,}000$ terms — no instrument can verify a claim about all $n$. The readout is exact for these four specimens because their distance to the limit moves monotonically, but for a general sequence a finite scan proves nothing; only the definition does.
Source: the $\varepsilon$–$N$ definition of a sequence limit — Limit of a sequence, with its history and formal statement.
At $\varepsilon = 0.05$ the stat for $1/n$ reads safe from $n = 21$. What exactly does that claim?
Keep $a_n = 1/n$ and set $\varepsilon = 0.25$. What is the first safe $n$?
Transfer: $b_n = 5 + (-0.8)^n$ swings above and below $5$ on alternating steps. What does it do as $n \to \infty$?
Convergence is a promise about tails: pick any tolerance, and past some index the sequence never leaves it. Oscillation is irrelevant, closeness is everything — and the required index $N$ is allowed to grow as the band narrows.