$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$

## Sequences

### Basic definitions

Definition of Sequence

Definition of Limit

Limit of a constant sequence

Limit example 1

Limit example 2 – Using less than or equal

Limit example 3 – Using the triangle inequality

Limit example 4

Non Convergent Examples

Common Mistakes

### General results

Limits are unique

The sequence $x^n$

The sequence $(|a_n|)$

The limit of $n^{-p}$ for $p$ a natural number

### Key techniques for showing convergence

Algebra of Limits for Sequences

Squeeze Rule

Bounded and monotonic implies convergence:

Bounded sequences

Monotonic sequences

Bounded and monotonic implies convergence

Extended Example

### Some sites:

If you are looking for pre-university level information on sequences and series, then have a look atÂ the PurpleMath pages.If you want a few pages on sequences and series that have some glaring errors then look at SparkNotes. (Their summary of divergent series is wrong for example, consider the sum 1-1+1-1+1-1+…)

Of course, you probably will have seen the Wikipedia pages on sequences and series.

The Khan Academy has a number of pages related to sequences and series.

And here’s a video

### Some books:

Here are some books about sequences and series that I like.*Sequential Introduction To Real Analysis* by JM Speight (Amazon UK) Full disclosure: The author is a colleague of mine!

*Numbers, Sequences and Series* by KE Hirst. (Amazon UK, Amazon US)

*Sequences and Series* by John A. Green (Amazon UK -from a penny!, Amazon US – a buck fifty!)

*Theory and Application of Infinite Series* by Konrad Knopp (Amazon UK, Amazon US)

### Anything else you would like to see?

If there is anything else you would like to see, then leave a comment below.