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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 𝑦 = 1 𝑥 in the figure, we define 𝑎 4 to be the area that is shaded. This gives a term of the sequence 𝑎 𝑛 .
Using an integral, give an exact expression for 𝑎 𝑛 .
The sequence 𝑎 𝑛 is clearly increasing. What does rectangle 𝐴 𝐵 𝐶 𝐷 tell us about the size of 𝑎 4 ?
What, therefore, can you give as an upper bound on all the 𝑎 𝑛 ?
What can you conclude about the sequence 𝑎 𝑛 ?
Q2:
Let 𝑁 ( 𝑥 ) = 𝑥 + 3 2 𝑥 .
Define 𝑁 ( 𝑥 ) = 𝑁 ( 𝑥 ) rounded to 6 decimal places. Now let 𝑥 = 1 , 𝑥 = 𝑁 ( 𝑥 ) = 2 . 0 0 0 0 0 0 , 𝑥 = 𝑁 ( 𝑥 ) = 1 . 7 5 0 0 0 0 , and so on. The sequence { 𝑥 } is eventually constant. At what value is this?
With 𝑁 ( 𝑥 ) = 𝑁 ( 𝑥 ) rounded to 10 decimal places, what is the limit, as 𝑛 → ∞ , of the sequence given by 𝑥 = 1 and 𝑥 = 𝑁 ( 𝑥 ) for 𝑛 ≥ 1 ?
If 𝑇 → 𝑧 as 𝑛 → ∞ , then, by continuity of 𝑁 , @ 𝑇 = 𝑁 ( 𝑇 ) → 𝑁 ( 𝑧 ) . So 𝑁 ( 𝑧 ) = 𝑧 . What would this 𝑧 be?
Q3:
Using induction, show that the sequence 1 , 2 , √ 7 , 3 √ 7 + 1 , 3 3 √ 7 + 1 + 1 , … is increasing and bounded, and find the limit of the sequence.
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