key term - (-1)^(n-1)

Definition

The expression (-1)^(n-1) is a mathematical term that appears in the context of alternating series. It represents a function that alternates between positive and negative values as the variable n changes. This term is a crucial component in determining the behavior and convergence of alternating series.

5 Must Know Facts For Your Next Test

1. The expression (-1)^(n-1) evaluates to 1 when n is odd and -1 when n is even, creating the alternating pattern of positive and negative terms in the series.
2. The sign of (-1)^(n-1) determines whether the term in the alternating series is positive or negative, which is crucial for determining the convergence or divergence of the series.
3. Alternating series that satisfy the Alternating Series Test (where the terms approach 0 and the series is absolutely convergent) can be evaluated using the formula: $\sum_{n=1}^\infty (-1)^{n-1}a_n$.
4. The behavior of (-1)^(n-1) is important in the Alternating Series Estimation Theorem, which provides a way to estimate the error in the partial sums of an alternating series.
5. The properties of (-1)^(n-1) are also used in the analysis of Fourier series, where the alternating sign is a key component in the representation of periodic functions.

Review Questions

• Explain how the expression (-1)^(n-1) contributes to the behavior of an alternating series.
  • The expression (-1)^(n-1) is crucial in determining the behavior of an alternating series. It causes the terms in the series to alternate between positive and negative values as the variable n changes. This alternating sign is a defining characteristic of an alternating series and is essential for the series to converge. The sign of (-1)^(n-1) determines whether the term in the series is positive or negative, which affects the sum of the series and its convergence or divergence.
• Describe how the properties of (-1)^(n-1) are used in the Alternating Series Estimation Theorem.
  • The Alternating Series Estimation Theorem provides a way to estimate the error in the partial sums of an alternating series. This theorem relies on the properties of (-1)^(n-1) to establish the convergence of the series and provide a bound on the error. Specifically, the theorem states that if the terms of an alternating series approach 0 and the series is absolutely convergent, then the error in the nth partial sum is bounded by the absolute value of the (n+1)th term, which is given by $|a_{n+1}|$. The alternating sign of (-1)^(n-1) is a crucial component in this theorem, as it ensures that the partial sums of the series provide a good approximation of the infinite sum.
• Analyze the role of (-1)^(n-1) in the representation of periodic functions using Fourier series.
  • The properties of (-1)^(n-1) are also important in the analysis of Fourier series, which are used to represent periodic functions. In a Fourier series, the coefficients of the sine and cosine terms alternate in sign, and this alternating sign is directly related to the expression (-1)^(n-1). Specifically, the Fourier series representation of a periodic function $f(x)$ is given by $\frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos(nx) + b_n\sin(nx)$, where the coefficients $a_n$ and $b_n$ are determined by the Fourier series formulas. The alternating sign of (-1)^(n-1) is a key component in these formulas, as it ensures the proper representation of the periodic function and its properties.

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 sum of the terms approaches a finite value as the number of terms increases.

Absolute Convergence:

Absolute convergence is a stronger form of convergence where the series formed by the absolute values of the terms also converges.