A sequence is a list of numbers, called terms, that follow a specific pattern or rule. Each term in a sequence is obtained by applying the same rule to the previous term.
To determine the limit of a sequence, you can use the formal definition of a limit, which states that a limit L is obtained when the terms of the sequence get closer and closer to L as the index of the terms gets larger and larger. You can also use various techniques such as the squeeze theorem and the ratio test to determine the limit of a sequence.
A convergent sequence is a sequence in which the terms approach a specific value as the index of the terms increases. In other words, the limit of a convergent sequence exists.
A divergent sequence is a sequence in which the terms do not approach a specific value as the index of the terms increases. In other words, the limit of a divergent sequence does not exist.
Exploring limits of sequences can be applied in various fields such as physics, engineering, and economics. For example, in physics, the concept of limits of sequences is used to study the behavior of systems over time. In economics, it can be used to analyze the growth of populations or the behavior of stock prices. In engineering, limits of sequences can be used to analyze the stability and performance of systems.