(-1)^n

Definition

The expression (-1)^n is a mathematical notation that represents the alternating sequence of positive and negative values, where the exponent 'n' determines the sign of the term. This expression is particularly relevant in the context of Alternating Series, a fundamental concept in Calculus II.

5 Must Know Facts For Your Next Test

1. The expression (-1)^n generates a sequence of values that alternates between 1 and -1, depending on the value of the exponent 'n'.
2. When 'n' is an even integer, the expression (-1)^n evaluates to 1, indicating a positive term in the series.
3. When 'n' is an odd integer, the expression (-1)^n evaluates to -1, indicating a negative term in the series.
4. The alternating nature of the terms in an Alternating Series is often a key factor in determining the convergence or divergence of the series.
5. Alternating Series that satisfy the Alternating Series Test can be guaranteed to converge, with the sum of the series being equal to the limit of the partial sums.

Review Questions

• Explain how the expression (-1)^n determines the sign of the terms in an Alternating Series.
  • The expression (-1)^n determines the sign of the terms in an Alternating Series by generating a sequence of values that alternates between 1 and -1. When 'n' is an even integer, the expression evaluates to 1, indicating a positive term in the series. When 'n' is an odd integer, the expression evaluates to -1, indicating a negative term in the series. This alternating pattern of positive and negative terms is a defining characteristic of Alternating Series and is crucial in understanding their convergence or divergence.
• Describe the relationship between the expression (-1)^n and the Alternating Series Test.
  • The expression (-1)^n is closely related to the Alternating Series Test, which is used to determine the convergence or divergence of Alternating Series. The Alternating Series Test states that if the absolute value of the terms in an Alternating Series decreases monotonically and the limit of the terms approaches 0, then the series converges. The alternating nature of the terms, as determined by the expression (-1)^n, is a key factor in the Alternating Series Test, as it ensures that the series satisfies the necessary conditions for convergence.
• Analyze how the expression (-1)^n can be used to identify the behavior of an Alternating Series in terms of absolute convergence.
  • The expression (-1)^n can be used to determine the absolute convergence of an Alternating Series. If the series formed by taking the absolute values of the terms in the original Alternating Series converges, then the original series is said to be absolutely convergent. This can be assessed by considering the series formed by the absolute values of the terms, which would not exhibit the alternating pattern generated by (-1)^n. If the absolute value series converges, then the original Alternating Series is absolutely convergent, and the sum of the series is equal to the limit of the partial sums. This relationship between (-1)^n and absolute convergence is a crucial concept in understanding the behavior of Alternating Series.

Related terms

Alternating Series: An alternating series is a series where the terms alternate in sign, typically between positive and negative values.

Convergence: Convergence refers to the behavior of a series where the partial sums approach a finite limit as the number of terms increases.

Absolute Convergence: Absolute convergence occurs when the series formed by taking the absolute values of the terms in the original series converges.