Question

Null sequence

Answer 1

[AI-generated answer — Bot]

A null sequence is a sequence of numbers that approaches zero as the index increases without ever becoming zero. In other words, for any positive number ε, there exists a point in the sequence after which all terms are less than ε in absolute value. Formally, a sequence {a_n} is said to be a null sequence if for every ε > 0, there exists an N such that |a_n| < ε for all n > N.

For example, the sequence {1/n} is a null sequence since as n gets larger, the terms 1/n get smaller and approach zero, but never actually become zero. Similarly, the sequence {sin(n)/n} is also a null sequence since sin(n) oscillates between -1 and 1 but is divided by n, which grows arbitrarily large, causing the entire sequence to approach zero.

Null sequences are important in mathematical analysis and the theory of limits because they provide a way to understand the behavior of functions as they approach a limit without actually reaching it.