Lesson: Convergent and Divergent Sequences Glencoe â€ĸ Precalculus â€ĸ Common Core

Mathematics

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

Worksheet: 6 Questions

Q1:

Using the graph of đ‘Ļ=1đ‘Ĩ in the figure, we define 𝑎īŠĒ to be the area that is shaded. This gives a term of the sequence 𝑎īŠ.

Using an integral, give an exact expression for 𝑎īŠ.

The sequence 𝑎īŠ is clearly increasing. What does rectangle 𝐴đĩđļ𝐷 tell us about the size of 𝑎īŠĒ?

What, therefore, can you give as an upper bound on all the 𝑎īŠ?

What can you conclude about the sequence 𝑎īŠ?

Q2:

Let 𝑁(đ‘Ĩ)=đ‘Ĩ+32đ‘ĨīŠ¨.

Define 𝑁(đ‘Ĩ)=𝑁(đ‘Ĩ)īŠ§ rounded to 6 decimal places. Now let đ‘Ĩ=1īŠ§, đ‘Ĩ=𝑁(đ‘Ĩ)=2.000000īŠ¨īŠ§īŠ§, đ‘Ĩ=𝑁(đ‘Ĩ)=1.750000īŠŠīŠ§īŠ¨, and so on. The sequence {đ‘Ĩ}īŠ is eventually constant. At what value is this?

With 𝑁(đ‘Ĩ)=𝑁(đ‘Ĩ)īŠ¨ rounded to 10 decimal places, what is the limit, as 𝑛→∞, of the sequence given by đ‘Ĩ=1īŠ§ and đ‘Ĩ=𝑁(đ‘Ĩ)īŠīŠ°īŠ§īŠ¨īŠ for 𝑛â‰Ĩ1?

If 𝑎→𝑧īŠ as 𝑛→∞, then, by continuity of 𝑁, 𝑎=𝑁(𝑎)→𝑁(𝑧)īŠīŠ°īŠ§īŠ. So 𝑁(𝑧)=𝑧. What would this 𝑧 be?

Q3:

Using induction, show that the sequence 1,2,√7,ī„3√7+1,ī„ž3ī„3√7+1+1,â€Ļ is increasing and bounded, and find the limit of the sequence.

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