In this lesson, we will learn how to determine whether a sequence is convergent or divergent.
Students will be able to
Q1:
Using the graph of π¦=1π₯ in the figure, we define ποͺ to be the area that is shaded. This gives a term of the sequence πο.
Using an integral, give an exact expression for πο.
The sequence πο is clearly increasing. What does rectangle π΄π΅πΆπ· tell us about the size of ποͺ?
What, therefore, can you give as an upper bound on all the πο?
What can you conclude about the sequence πο?
Q2:
Let π(π₯)=π₯+32π₯ο¨.
Define π(π₯)=π(π₯)ο§ rounded to 6 decimal places. Now let π₯=1ο§, π₯=π(π₯)=2.000000ο¨ο§ο§, π₯=π(π₯)=1.750000ο©ο§ο¨, and so on. The sequence {π₯}ο is eventually constant. At what value is this?
With π(π₯)=π(π₯)ο¨ rounded to 10 decimal places, what is the limit, as πββ, of the sequence given by π₯=1ο§ and π₯=π(π₯)οο°ο§ο¨ο for πβ₯1?
If πβπ§ο as πββ, then, by continuity of π, π=π(π)βπ(π§)οο°ο§ο. So π(π§)=π§. What would this π§ be?
Q3:
Using induction, show that the sequence 1,2,β7,ο3β7+1,ο3ο3β7+1+1,β¦ is increasing and bounded, and find the limit of the sequence.
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