Limit of a Recurrence Relation

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Limit of a Recurrence Relation

• Recurrence Relations

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The limit of a recurrence relation
Consider the recurrence relation
un1 0.6un 8 , with uo 5 ---
(1)
Using the graphic calculator we can easily
generate the terms of the sequence obtained
from the above recurrence relation by using the
procedure 5 , ENTER , , 0.6 , , 8 ,
ENTER , ENTER , ENTER , - - which leads to
5 , 11 , 14.6 , 16.76 , 18.056 , - - - - - -
If you continue long enough with the graphic
calculator then you will see that the sequence
gets closer and closer and closer to the number
20. We call this the limit of the sequence and
we write L 20
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We can determine this limit algebraically as
follows. Let L be the limit of the recurrence
relation (1) above. Then L 0.6L 8 L 0.6L
8 0.4L 8 L 8/0.4 20
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• In each example below
• use the graphic calculator to generate a
• sequence and find the limit.
• Obtain the limit algebraically.
• 1. un1 0.8un 5 , with uo 12
• 2. un1 0.1un 4 , with uo 5
• 3. un1 0.5un 8 , with uo 1
• 4. un1 -0.2un 7 , with uo 10
• 5. un1 -0.75un 2 , with uo 20
• 6. un1 0.95un 3 , with uo 5
• 7. un1 -0.6un 2 , with uo 12
• 8. un1 5 - 0.3un , with uo 1
• 9. un1 0.64un 10 , with uo 10
• 10. un1 0.82un 6 , with uo 50
• 11. un1 0.3un 12 , with uo 15
• 12. un1 4 - 0.85un , with uo 1