Sequences and series

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


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) 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.

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