Divergent sequence

Definition

A divergent sequence is a sequence of numbers that does not converge to a specific limit as the number of terms approaches infinity. Instead, the terms may increase or decrease without bound, oscillate, or exhibit some other form of unbounded behavior. Understanding divergent sequences is crucial for grasping the broader concepts of limits and infinite series.

5 Must Know Facts For Your Next Test

1. A sequence diverges if it does not settle down to a single value as more terms are added, meaning its limit does not exist.
2. Common examples of divergent sequences include the sequence of natural numbers and alternating sequences like 1, -1, 1, -1, ...
3. Divergent sequences can either diverge to infinity or negative infinity, or oscillate without approaching any single value.
4. Determining whether a sequence is divergent can often be done using tests such as the n-th term test for divergence.
5. Divergence is significant in calculus because it helps in understanding which series can be summed and which cannot.

Review Questions

• How can you determine if a sequence is divergent, and what tests can you apply?
  • To determine if a sequence is divergent, one can analyze its behavior as the number of terms approaches infinity. If the terms do not approach a specific limit, it indicates divergence. Common tests include the n-th term test for divergence, where if the limit of the terms does not equal zero, the sequence diverges. Other methods involve comparing with known divergent sequences to understand their behavior.
• Explain how divergent sequences relate to infinite series and their convergence tests.
  • Divergent sequences are closely linked to infinite series because an infinite series can only be summed if its corresponding sequence converges. When evaluating an infinite series, if the sequence of its partial sums diverges, then the series itself is said to diverge. This relationship emphasizes the importance of recognizing divergent sequences in order to apply appropriate convergence tests, such as the ratio test or root test, to assess whether an infinite series converges or diverges.
• Evaluate the implications of having a divergent sequence in real-world scenarios and mathematical applications.
  • The presence of a divergent sequence in real-world contexts can have significant implications, especially in fields like physics or economics where certain processes are modeled using sequences. For example, in financial models where investments grow without bounds, understanding divergence is critical for assessing risk and stability. Mathematically, recognizing divergence helps prevent erroneous conclusions when attempting to sum infinite processes or analyze long-term behaviors. In essence, divergent sequences serve as warning signs that certain assumptions about stability and predictability may not hold.

Related terms

Convergent Sequence: A convergent sequence is a sequence that approaches a specific finite limit as the number of terms increases indefinitely.

Limit of a Sequence: The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity, which may be finite or infinite.

Infinite Series: An infinite series is the sum of the terms of an infinite sequence, which may converge to a finite value or diverge.