# Sequence limit

A sequence is a list of numbers a_1, a_2, a_3, and so on. Its limit is the value the terms approach as n gets very large.

Maths (university) · Part 07: Sequences and series · Card 1 of 9 · Last updated 2026-10-19

## Analogy

Walking towards a wall by halving the distance each step: you get closer and closer but never pass it.

## Example

**a_n = (2n + 1) / n**

n = 1: 3. n = 10: 2.1.
n = 100: 2.01.
Write as 2 + 1/n.
Limit is 2.

## In practice

Sequence limits are the starting point for testing whether an infinite sum has a total.

## Practice

### Flashcards

- **What is the limit of the sequence (2n + 1)/n?** 2.

## Questions people ask

### What does it mean for a sequence to converge?

A sequence converges if its terms approach one fixed number as n grows. If they grow without bound or keep jumping around without settling, it diverges. The sequence 1, -1, 1, -1 and so on diverges, since it never settles.

### How is a sequence different from a series?

A sequence is a list of numbers. A series is the sum of those numbers. The sequence 1, 1/2, 1/4, and so on has limit 0, but the series 1 + 1/2 + 1/4 + and so on adds up to 2. They answer different questions.

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