Types of Divergence

Real Analysis · Axiom Academy

A sequence can fail to converge in exactly three ways — running off to , off to , or bouncing forever like a_n=(-1)^n . 1. The Bar Every Sequence Must Clear To converge to L , a sequence must eventually be trapped : pick any tolerance , and from some index N onward all terms satisfy . The animation draws that -band around a candidate L and lets the terms try to settle into it. Take a_n=n : Each term tops the last and the sequence climbs without bound . Drop an -band at any finite candidate L — the terms eventually rise above it and never return. The animation marches the terms up the number line; the candidate band scrolls by underneath, left behind for good. For any finite L and any , every term with satisfies , so it falls outside the band — infinitely many terms escape. We write , but this is divergence : is not a real number, so there is no limit L to converge to. The mirror image: a_n=-n gives Now the terms sink below every floor . Same failure, opposite direction — the animation marches the terms down the line, past any candidate band, never coming back up. For any finite L and any , every term with satisfies , dropping below the band. We write . Like Mode 1, the terms run off the end of the real line, so no finite limit exists — it diverges.

This is the written version of the interactive lesson above. See the full Real Analysis course.