Iv 4 limits of sequences introduction
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IV.4 Limits of sequences introduction. What happens to a sequence a n when n tends to infinity? A sequence can have one of three types of behaviour: Convergence Divergence Neither of the above…. Graphical interpretation of limits for explicit sequences. Examples of converging sequences.

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IV.4 Limits of sequences introduction

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Iv 4 limits of sequences introduction

IV.4 Limits of sequencesintroduction


Iv 4 limits of sequences introduction

  • What happens to a sequence an when n tends to infinity?

  • A sequence can have one of three types of behaviour:

    • Convergence

    • Divergence

    • Neither of the above…


Graphical interpretation of limits for explicit sequences

Graphical interpretation of limits for explicit sequences


Examples of converging sequences

Examples of converging sequences

The sequence converges to a unique value: 0

We write this as

limn  an = 0


Examples of converging sequences1

Examples of converging sequences

The sequence converges to a unique value: 1

We write this as

limn  an = 1


Examples of diverging sequences

Examples of diverging sequences

The sequence diverges, it has no real limit.

We can write this as

limn  an = + 


Examples of diverging sequences1

Examples of diverging sequences

The sequence diverges; it has no real limit.


A sequence that neither converges nor diverges

A sequence that neither converges nor diverges

This sequence neither converges nor diverges.

In this particular case, it is periodic.


Graphical interpretation of limits for implicit sequences

Graphical interpretation of limits for implicit sequences


A recursion that converges

A recursion that converges

This sequence is defined by an implicit formula

an+1 = f(an)

From the cobweb diagram, we see that it converges to 0:

We write

limn  an = 0


A recursion that converges1

A recursion that converges

This sequence is defined by an implicit formula

an+1 = f(an)

From the cobweb diagram, we see that it converges to a value a:

We write

limn  an = a


A recursion that converges2

A recursion that converges

When the sequence converges, it always converges to

A point at the intersection of the an+1 = an line (dashed), and

the an+1 = f(an) curve (solid)


A recursion that converges3

A recursion that converges

These points are called fixed points

Fixed points satisfy the equation an+1 = an = f(an)


A recursion that diverges

A recursion that diverges

This sequence diverges.

We can write

limn an= 


A recursion that neither converges nor diverges

A recursion that neither converges nor diverges

This sequence neither converges not diverges.

It is periodic.

The fixed points exists but they are not the limit of the sequence


A recursion that neither converges nor diverges1

A recursion that neither converges nor diverges

This sequence neither converges not diverges.

It is chaotic.

The fixed points exists but they are not the limit of the sequence


Not all fixed points are limits

Not all fixed points are limits


Not all fixed points are limits1

Not all fixed points are limits


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