Mathematical Analysis

Sequences and Series

When is a sequence monotonic?

A sequence is called monotonic if one of the following properties hold: In the sequence {xn}n≥1, it is increasing if and only if xn <>n + 1 for any n ≥ 1, or it is decreasing if and only if xn > xn + 1 for any n ≥ 1.

For example, in order to check that the sequence {2n}n≥1 is increasing: Let n ≥ 1; that gives 2n+1 = 2n 2. Because 2 is greater than 1, which means that 1 × 2n < 2="" ×="">n; thus 2n <>n + 1, which shows the sequence is increasing.


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