Help

Pick a sequence from the dropdown. Sequences fall into two groups: those that converge to a constant (limit 0), and those that diverge to +∞.

In User mode, set ε (for convergent sequences) or B (for divergent ones) yourself with the box or slider, then try to find an N that works: click the point on the plot you think works, or type a value into the "Your N" box directly. Points from N onward turn green if they satisfy the condition and red if they don't — watch the colours update live as you change N.

In Challenge mode, three fixed values are given (increasingly strict). Find an N for each, then press "Check" to test your answer.

Use the Zoom n and Zoom aₙ controls to zoom either axis in or out by a factor of two. The n-axis always starts at 0; the aₙ-axis is always centred on 0. Press "Reset view" to return to the default framing for the current sequence and ε/B. The number shown near the top of the plot is the current vertical scale; the number near the bottom right is the largest n currently visible.

"How to find N" shows a general written method for finding a workable (not necessarily optimal) N. "Proof" shows the formal argument that it works.

Overview

A sequence (aₙ) converges to a limit L if, for every ε > 0, there is some N such that n > N guarantees |aₙ − L| < ε. No matter how small the value of ε it is always possible to find an N such that all later sequence elements lie in the ε-neighbourhood of L.

A sequence diverges to +∞ if, for every B > 0, there is some N such that n > N guarantees aₙ > B. However large B is, it is possible to find an N such that all later terms stay above B.

Proving either property means producing a working N for an arbitrary ε or B, along with a chain of inequalities showing that N actually does the job.

How to find N