Part 1/5:

Understanding Monotonic Sequences

Monotonic sequences are a fundamental concept in mathematics, particularly in the study of real analysis. They are classified based on whether they are consistently increasing or decreasing. This article provides a comprehensive overview and analysis of monotonic sequences, building on their definitions and exploring specific examples.

Definition of Monotonic Sequences

A sequence ( a_n ) is defined as increasing if every term is less than the subsequent term, which mathematically can be expressed as:

[ a_n < a_{n+1} \quad \text{for all } n \geq 1 ]

For instance, if we have a sequence where ( a_1 < a_2 < a_3 < a_4 < \ldots ), this sequence is categorized as increasing.