Convergent and Divergent Sequences - Limits of Sequences - Calculus

In mathematics, sequences are ordered lists of numbers. The behaviour of a sequence as the terms increase (approach infinity) determines whether it is convergent or divergent.
1. Convergent Sequences

A sequence is convergent if its terms approach a single finite value as the sequence progresses toward infinity.

2. Divergent Sequences

A sequence is divergent if it does not converge to any finite limit. Divergence can occur in several ways:

The sequence grows without bound.

The sequence oscillates without settling near a single value.

The sequence fails to approach any specific finite value.

Key Types of Divergence:
Unbounded Divergence: The terms grow infinitely large in magnitude.
Example: a_n=n, which diverges to infinity.
Oscillatory Divergence: The terms oscillate without settling.
Example: a_n}=(−1)^n, which alternates between 1 and −1.