In this lesson, we will learn how to use the monotone convergence theorem to test for convergence.

Q1:

Which of the following describes the least upper bound ℓ of a subset 𝑆 of numbers?

Q2:

Suppose 𝑆 is a subset of real numbers that has an upper bound 𝑏.

Suppose that ℓ is a least upper bound of 𝑆. Which of the following relations between ℓ and 𝑏 must be true?

Suppose ℓ is a second least upper bound for 𝑆. Which of the following relations between ℓ and ℓ must be true?

Q3:

Consider the sequence 𝑎=3𝑛−42𝑛−1 for 𝑛≥1.

Is 𝑎>𝑎?

Is 𝑎>𝑎?

Define 𝑃,𝑄,𝑅, and 𝑆 by 𝑎=𝑃𝑄 and 𝑎=𝑅𝑆. Write 𝑃𝑆−𝑄𝑅 in simplified form.

Using the above and the quadratic formula, find the smallest integer 𝑁 so that 𝑎>𝑎 whenever 𝑛≥𝑁.

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