Final answer:
To prove a sequence is divergent, one can show that there is no limit to which the terms of the sequence approach as n increases or that the terms do not tend to zero, exemplified by a sequence like aₙ = n which clearly diverges as it increases without bound.
Step-by-step explanation:
To prove a sequence is divergent, one must use the definition of divergence. A sequence diverges if it does not converge to a single value as n approaches infinity.
That is, for a sequence to be divergent, given any number L, you can find a place in the sequence after which the sequence does not stay arbitrarily close to L. One of the methods to show divergence is to prove that the limit of the sequence as n approaches infinity does not exist or is not finite.
When working with sequences, the limit of a sequence is a foundational concept. If the terms of a sequence get arbitrarily close to a fixed value as n gets larger, the sequence is said to converge to that value.
Conversely, if there is no single value that the terms of the sequence approach as n increases indefinitely, or if the terms continue to grow without bound, the sequence is considered to be divergent.
An example would be the sequence aₙ = n, which is clearly divergent since it increases without bound. No matter what number you choose as L, you can always find terms of the sequence that are greater than L, so the sequence does not settle down to any particular value.
Another common way to show divergence is by using the nth term test for divergence: if the limit of an as n approaches infinity does not equal zero or if it does not exist, then the series ∑aₙ is divergent. This is critical because if a series converges, the terms in the series must be getting smaller and smaller, specifically tending towards zero.