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