In this section, we will introduce the idea of an infinite sequence and what it means to say one converges or diverges.

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Presentation transcript:

In this section, we will introduce the idea of an infinite sequence and what it means to say one converges or diverges.

For example:

The index variable can be anything; all mean the same thing. The starting index is insignificant. The first term may be We will be interested in the long–run behavior.

Determine whether each of the following sequences converge.

Show that the sequenceconverges to 1.

Suppose the sequence is nondecreasing and bounded above by A. Then converges to some value ≤ A. Suppose the sequence is nonincreasing and bounded below by B. Then converges to some value ≥ B.

Consider defined by a 1 = 1 and. Show that converges and the limit is ≤ 4.