Alternating Sequence

Definition

An alternating sequence is a sequence of numbers in which the terms alternately take positive and negative values. This characteristic leads to interesting properties regarding convergence and boundedness, particularly in mathematical analysis, where understanding the behavior of sequences is crucial for determining limits and their overall properties.

5 Must Know Facts For Your Next Test

1. An alternating sequence can be expressed in the form $a_n = (-1)^n b_n$, where $b_n$ is a positive sequence.
2. If an alternating sequence's absolute terms $b_n$ decrease to zero, the sequence converges by the Alternating Series Test.
3. An example of an alternating sequence is $1, -1, 1, -1, ext{...}$ which does not converge.
4. The convergence of an alternating sequence may depend on the rate at which its absolute terms approach zero.
5. Alternating sequences can often be used to illustrate concepts like conditional convergence, where the series converges but not absolutely.

Review Questions

• How does the Alternating Series Test determine the convergence of an alternating sequence?
  • The Alternating Series Test states that if the absolute terms of an alternating sequence decrease monotonically to zero, then the series converges. Specifically, for a sequence defined as $a_n = (-1)^n b_n$, if $b_n$ is positive, monotonically decreasing, and $ ext{lim}_{n \to \infty} b_n = 0$, then the series formed by this sequence converges. This test helps in analyzing the behavior of sequences that alternate in sign.
• Contrast an alternating sequence with a monotonic sequence regarding their convergence properties.
  • An alternating sequence fluctuates between positive and negative values, while a monotonic sequence consistently increases or decreases without changing direction. Although an alternating sequence can converge under certain conditions (like diminishing absolute terms), a monotonic sequence that is bounded must converge due to the Monotone Convergence Theorem. Therefore, while both can exhibit convergence, they do so under different rules and conditions.
• Evaluate how the concept of conditional convergence relates to alternating sequences and their absolute counterparts.
  • Conditional convergence occurs when an alternating series converges, but the series of its absolute values diverges. This is particularly relevant for alternating sequences because they often display this behavior. For instance, the series formed by $(-1)^n/n$ converges conditionally, while the series of $1/n$ diverges. Understanding this distinction is crucial when studying sequences and series in mathematical analysis as it highlights different types of convergence and their implications for infinite sums.

Related terms

Convergence: The property of a sequence whereby its terms approach a specific value as the index increases indefinitely.

Bounded Sequence: A sequence is bounded if there exists a real number that serves as an upper limit and another that serves as a lower limit for all its terms.

Monotonic Sequence: A sequence that is either entirely non-increasing or non-decreasing, which provides insights into its convergence behavior.