# Sequences and series

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

## Sequences

### 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 at  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)