The divergence criteria are a set of conditions that determine whether a sequence or series converges or diverges. These criteria involve comparing the beha...
The divergence criteria are a set of conditions that determine whether a sequence or series converges or diverges. These criteria involve comparing the behavior of the sequence or series to that of a related sequence or series. If the two sequences exhibit similar behavior, then the original sequence is said to converge. However, if they exhibit different behaviors, then the original sequence diverges.
A sequence or series converges if its limit exists and is finite. This means that the sequence approaches a single specific value as the number of terms increases without bound.
A sequence or series diverges if its limit is either infinity or undefined. This means that the sequence approaches infinity or oscillates between positive and negative infinity as the number of terms increases without bound.
The sequential criterion is a specific divergence criterion that applies to sequences of real numbers. It states that a sequence of real numbers converges if and only if its limit is a real number and the sequence converges to that limit.
For example, consider the sequence (a_n = \frac{1}{n}). As (n) increases, the value of (a_n) approaches 0. Therefore, the sequence converges to 0 according to the sequential criterion.
On the other hand, consider the sequence (a_n = (-1)^n). As (n) increases, the value of (a_n) alternates between 1 and -1. Therefore, the sequence diverges according to the sequential criterion