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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slide2
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…
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.

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

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