In this lesson, we will learn how to determine whether a sequence is convergent or divergent.
Q1:
Using the graph of 𝑦=1𝑥 in the figure, we define 𝑎 to be the area that is shaded. This gives a term of the sequence 𝑎.
Using an integral, give an exact expression for 𝑎.
The sequence 𝑎 is clearly increasing. What does rectangle 𝐴𝐵𝐶𝐷 tell us about the size of 𝑎?
What, therefore, can you give as an upper bound on all the 𝑎?
What can you conclude about the sequence 𝑎?
Q2:
Let 𝑁(𝑥)=𝑥+32𝑥.
Define 𝑁(𝑥)=𝑁(𝑥) rounded to 6 decimal places. Now let 𝑥=1, 𝑥=𝑁(𝑥)=2.000000, 𝑥=𝑁(𝑥)=1.750000, 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 𝑎→𝑧 as 𝑛→∞, then, by continuity of 𝑁, 𝑎=𝑁(𝑎)→𝑁(𝑧). So 𝑁(𝑧)=𝑧. What would this 𝑧 be?
Q3:
Using induction, show that the sequence 1,2,√7,3√7+1,33√7+1+1,… is increasing and bounded, and find the limit of the sequence.
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