Limits of sequences

Distinguish convergence from divergence and explain limits and their calculation conditions.

Meaning

A sequence limit is a value that terms approach as their indices grow without bound. Convergence means that all sufficiently late terms can be made as close to that value as desired.

Example

The terms of are 1, … and decrease. Every term is positive, but the terms approach 0 as closely as desired, so the limit is 0.To be within 0.01 of zero, take n>100; every later term also qualifies. For any smaller positive tolerance ε, n>1/ε works, showing that the limit is zero.A limit need not occur as an actual term. A sequence such as which alternates between two values, does not converge to one value.

Practice for this concept

Scope and references

Topics

Convergence, divergence, infinity, calculating limits

Learning goal

Distinguish convergence from divergence and explain limits and their calculation conditions.

Helpful background

Review terms of sequences and distances between real numbers.