Comparing Converging and Diverging Sequences

Every infinite sequence is either convergent or divergent. A convergent sequence has a limit — that is, it approaches a real number. A divergent sequence doesn’t have a limit.

Here’s an example of a convergent sequence:

image0.png

This sequence approaches 0, so:

image1.png

Thus, this sequence converges to 0.

Here’s another convergent sequence:

image2.png

This time, the sequence approaches 8 from above and below, so:

image3.png

In many cases, however, a sequence diverges — that is, it fails to approach any real number. Divergence can happen in two ways. The most obvious type of divergence occurs when a sequence explodes to infinity or negative infinity — that is, it gets farther and farther away from 0 with every term. Here are a few examples:

–1, –2, –3, –4, –5, –6, –7, . . .

ln 1, ln 2, ln 3, ln 4, ln 5, . . .

2, 3, 5, 7, 11, 13, 17, . . .

In each of these cases, the sequence approaches either

image4.png

so the limit of the sequence does not exist. Therefore, the sequence is divergent.

A second type of divergence occurs when a sequence oscillates between two or more values. For example:

0, 7, 0, 7, 0, 7, 0, 7, . . .

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, . . .

In these cases, the sequence bounces around indefinitely, never settling in on a value. Again, the limit of the sequence does not exist, so the sequence is divergent.

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