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In this lesson, we will learn how to determine whether a sequence is convergent or divergent.
Q1:
Using the graph of π¦ = 1 π₯ in the figure, we define π 4 to be the area that is shaded. This gives a term of the sequence π π .
Using an integral, give an exact expression for π π .
The sequence π π is clearly increasing. What does rectangle π΄ π΅ πΆ π· tell us about the size of π 4 ?
What, therefore, can you give as an upper bound on all the π π ?
What can you conclude about the sequence π π ?
Q2:
Let π ( π₯ ) = π₯ + 3 2 π₯ ο¨ .
Define π ( π₯ ) = π ( π₯ ) ο§ rounded to 6 decimal places. Now let π₯ = 1 ο§ , π₯ = π ( π₯ ) = 2 . 0 0 0 0 0 0 ο¨ ο§ ο§ , π₯ = π ( π₯ ) = 1 . 7 5 0 0 0 0 ο© 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 π β β , of the sequence given by π₯ = 1 ο§ and π₯ = π ( π₯ ) ο ο° ο§ ο¨ ο for π β₯ 1 ?
If π β π§ ο as π β β , then, by continuity of π , π = π ( π ) β π ( π§ ) ο ο° ο§ ο . So π ( π§ ) = π§ . What would this π§ be?
Q3:
Using induction, show that the sequence 1 , 2 , β 7 , ο 3 β 7 + 1 , ο 3 ο 3 β 7 + 1 + 1 , β¦ is increasing and bounded, and find the limit of the sequence.
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