Lecture 14:

Here comes sequences!

Sequences

Definition. A sequence of real numbers (or a sequence in \(\R\)) is a function \(X:\N\to\R\).

 

Example. Let \(X:\N\to\R\) be given by \(X(n) = \dfrac{n}{n+1}\).

So, \(X(1) = \frac{1}{2}\), \(X(2) = \frac{2}{3}\), \(X(3) = \frac{3}{4}\), and so on.

Defining Sequences

Instead of defining a sequence by saying something like

 

"Let \(X = \left(1-\dfrac{1}{n}\right)\)."

 

We will often write

 

"Let \(X = (x_{n})\) be a sequence where \(x_{n} = 1-\dfrac{1}{n}\)."

 

With this notation we have a name, \(X\), by which we can refer to the whole sequence, and we have a name, \(x_{n}\), for the \(n\)th term in the sequence.

Defining Sequences

We can define sequences by many means:

 

Pattern:

\[X = (7, 10, 13, \ldots)\]

\[Y = \left(\frac{1}{2},\frac{2}{5},\frac{3}{10},\frac{4}{17},\ldots\right)\]

Formula:

\[X = (3n+4 : n\in\N)\]

\[Y = \left(\frac{k}{k^2+1}\right)\]

Recursive:

\[X = (x_{n}), \text{ where }x_{1} = 7,\ x_{n} = x_{n-1}+3\text{ for }n\geq 2\]

\[Z = (z_{n}), \text{ where }z_{1}=1,\ z_{2} = 1\ z_{n}=z_{n-2}-z_{n-1}\text{ for }n\geq 3\]

Limits of sequences

Given a sequence \((x_{n})\), what does it mean to say that 

\[\lim_{n\to\infty}x_{n} = L?\]

 

Example. Consider the sequence \((\frac{1}{n})\). What do we mean when we say

\[\lim_{n\to\infty}\frac{1}{n} =  \]

Practice Problem:

 

Define the sequence \(X=(x_{n})\) as follows: 

\[x_{1} = 1,\quad x_{2} = 3,\quad\text{and}\quad x_{n} = x_{n-1}^{2}-x_{n-2}\text{ for }n\geq 3.\]

Find the \(4\)th term of the sequence.