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Lesson: Convergent and Divergent Sequences Mathematics • 12th Grade

In this lesson, we will learn how to determine whether a sequence is convergent or divergent.

Lesson Plan

Lesson Worksheet

Q1:

Using the graph of 𝑦=1π‘₯ in the figure, we define π‘ŽοŠͺ to be the area that is shaded. This gives a term of the sequence π‘ŽοŠ.

Using an integral, give an exact expression for π‘ŽοŠ.

The sequence π‘ŽοŠ is clearly increasing. What does rectangle 𝐴𝐡𝐢𝐷 tell us about the size of π‘ŽοŠͺ?

What, therefore, can you give as an upper bound on all the π‘ŽοŠ?

What can you conclude about the sequence π‘ŽοŠ?

Q2:

Let 𝑁(π‘₯)=π‘₯+32π‘₯.

Define 𝑁(π‘₯)=𝑁(π‘₯) rounded to 6 decimal places. Now let π‘₯=1, π‘₯=𝑁(π‘₯)=2.000000, π‘₯=𝑁(π‘₯)=1.750000, and so on. The sequence {π‘₯} is eventually constant. At what value is this?

With 𝑁(π‘₯)=𝑁(π‘₯) rounded to 10 decimal places, what is the limit, as π‘›β†’βˆž, of the sequence given by π‘₯=1 and π‘₯=𝑁(π‘₯) for 𝑛β‰₯1?

If π‘Žβ†’π‘§οŠ as π‘›β†’βˆž, then, by continuity of 𝑁, π‘Ž=𝑁(π‘Ž)→𝑁(𝑧). So 𝑁(𝑧)=𝑧. What would this 𝑧 be?

Q3:

Using induction, show that the sequence 1,2,√7,3√7+1,ο„ž33√7+1+1,… is increasing and bounded, and find the limit of the sequence.

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